Groups St Andrews 2017 in Birmingham (London Mathematical Society Lecture Note Series, Band 455) 110872874X, 9781108728744

Table of contents :
Contents
Introduction
Finite simple groups and fusion systems
Finite and infinite quotients of discrete and indiscrete groups
Local-global conjectures and blocks of finite simple groups
A survey on some methods of generating finite simple groups
One-relator groups: an overview
New progress in products of conjugacy classes in finite groups
Aspherical relative presentations all over again
Simple groups, generation and probabilistic methods
Irreducible subgroups of simple algebraic groups – a survey
Practical computation with linear groups over infinite domains
Beauville p-groups: a survey
Structural criteria in factorised groups via conjugacy class sizes
Growth in linear algebraic groups and permutation groups: towards a unified perspective
L2-Betti numbers and their analogues in positive characteristic
On the pronormality of subgroups of odd index in finite simple groups
Vertex stabilizers of graphs with primitive automorphism groups and a strong version of the Sims conjecture
On the character degrees of a Sylow p-subgroup of a finite Chevalley group G(pf ) over a bad prime
Patterns on symmetric Riemann surfaces
Subgroups of twisted wreath products
Some remarks on self-dual codes invariant under almost simple permutation groups
Test elements: from pro-p to discrete groups

Citation preview

LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor M. Reid, Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom The titles below are available from booksellers, or from Cambridge University Press at www.cambridge.org/mathematics 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400

Surveys in combinatorics 2007, A. HILTON & J. TALBOT (eds) Surveys in contemporary mathematics, N. YOUNG & Y. CHOI (eds) Transcendental dynamics and complex analysis, P.J. RIPPON & G.M. STALLARD (eds) Model theory with applications to algebra and analysis I, Z. CHATZIDAKIS, D. MACPHERSON, A. PILLAY & A. WILKIE (eds) Model theory with applications to algebra and analysis II, Z. CHATZIDAKIS, D. MACPHERSON, A. PILLAY & A. WILKIE (eds) Finite von Neumann algebras and masas, A.M. SINCLAIR & R.R. SMITH Number theory and polynomials, J. MCKEE & C. SMYTH (eds) Trends in stochastic analysis, J. BLATH, P. MÖRTERS & M. SCHEUTZOW (eds) Groups and analysis, K. TENT (ed) Non-equilibrium statistical mechanics and turbulence, J. CARDY, G. FALKOVICH & K. GAWEDZKI Elliptic curves and big Galois representations, D. DELBOURGO Algebraic theory of differential equations, M.A.H. MACCALLUM & A.V. MIKHAILOV (eds) Geometric and cohomological methods in group theory, M.R. BRIDSON, P.H. KROPHOLLER & I.J. LEARY (eds) Moduli spaces and vector bundles, L. BRAMBILA-PAZ, S.B. BRADLOW, O. GARCÍA-PRADA & S. RAMANAN (eds) Zariski geometries, B. ZILBER Words: Notes on verbal width in groups, D. SEGAL Differential tensor algebras and their module categories, R. BAUTISTA, L. SALMERÓN & R. ZUAZUA Foundations of computational mathematics, Hong Kong 2008, F. CUCKER, A. PINKUS & M.J. TODD (eds) Partial differential equations and fluid mechanics, J.C. ROBINSON & J.L. RODRIGO (eds) Surveys in combinatorics 2009, S. HUCZYNSKA, J.D. MITCHELL & C.M. RONEY-DOUGAL (eds) Highly oscillatory problems, B. ENGQUIST, A. FOKAS, E. HAIRER & A. ISERLES (eds) Random matrices: High dimensional phenomena, G. BLOWER Geometry of Riemann surfaces, F.P. GARDINER, G. GONZÁLEZ-DIEZ & C. KOUROUNIOTIS (eds) Epidemics and rumours in complex networks, M. DRAIEF & L. MASSOULIÉ Theory of p-adic distributions, S. ALBEVERIO, A.YU. KHRENNIKOV & V.M. SHELKOVICH ´ Conformal fractals, F. PRZYTYCKI & M. URBANSKI Moonshine: The first quarter century and beyond, J. LEPOWSKY, J. MCKAY & M.P. TUITE (eds) Smoothness, regularity and complete intersection, J. MAJADAS & A. G. RODICIO Geometric analysis of hyperbolic differential equations: An introduction, S. ALINHAC Triangulated categories, T. HOLM, P. JØRGENSEN & R. ROUQUIER (eds) Permutation patterns, S. LINTON, N. RUŠKUC & V. VATTER (eds) An introduction to Galois cohomology and its applications, G. BERHUY Probability and mathematical genetics, N. H. BINGHAM & C. M. GOLDIE (eds) Finite and algorithmic model theory, J. ESPARZA, C. MICHAUX & C. STEINHORN (eds) Real and complex singularities, M. MANOEL, M.C. ROMERO FUSTER & C.T.C WALL (eds) Symmetries and integrability of difference equations, D. LEVI, P. OLVER, Z. THOMOVA & P. WINTERNITZ (eds) ˇ Forcing with random variables and proof complexity, J. KRAJÍCEK Motivic integration and its interactions with model theory and non-Archimedean geometry I, R. CLUCKERS, J. NICAISE & J. SEBAG (eds) Motivic integration and its interactions with model theory and non-Archimedean geometry II, R. CLUCKERS, J. NICAISE & J. SEBAG (eds) Entropy of hidden Markov processes and connections to dynamical systems, B. MARCUS, K. PETERSEN & T. WEISSMAN (eds) Independence-friendly logic, A.L. MANN, G. SANDU & M. SEVENSTER Groups St Andrews 2009 in Bath I, C.M. CAMPBELL et al (eds) Groups St Andrews 2009 in Bath II, C.M. CAMPBELL et al (eds) Random fields on the sphere, D. MARINUCCI & G. PECCATI Localization in periodic potentials, D.E. PELINOVSKY Fusion systems in algebra and topology, M. ASCHBACHER, R. KESSAR & B. OLIVER Surveys in combinatorics 2011, R. CHAPMAN (ed) Non-abelian fundamental groups and Iwasawa theory, J. COATES et al (eds) Variational problems in differential geometry, R. BIELAWSKI, K. HOUSTON & M. SPEIGHT (eds) How groups grow, A. MANN Arithmetic differential operators over the p-adic integers, C.C. RALPH & S.R. SIMANCA Hyperbolic geometry and applications in quantum chaos and cosmology, J. BOLTE & F. STEINER (eds) Mathematical models in contact mechanics, M. SOFONEA & A. MATEI Circuit double cover of graphs, C.-Q. ZHANG Dense sphere packings: a blueprint for formal proofs, T. HALES

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A double Hall algebra approach to affine quantum Schur–Weyl theory, B. DENG, J. DU & Q. FU Mathematical aspects of fluid mechanics, J.C. ROBINSON, J.L. RODRIGO & W. SADOWSKI (eds) Foundations of computational mathematics, Budapest 2011, F. CUCKER, T. KRICK, A. PINKUS & A. SZANTO (eds) Operator methods for boundary value problems, S. HASSI, H.S.V. DE SNOO & F.H. SZAFRANIEC (eds) Torsors, étale homotopy and applications to rational points, A.N. SKOROBOGATOV (ed) Appalachian set theory, J. CUMMINGS & E. SCHIMMERLING (eds) The maximal subgroups of the low-dimensional finite classical groups, J.N. BRAY, D.F. HOLT & C.M. RONEY-DOUGAL Complexity science: the Warwick master’s course, R. BALL, V. KOLOKOLTSOV & R.S. MACKAY (eds) Surveys in combinatorics 2013, S.R. BLACKBURN, S. GERKE & M. WILDON (eds) Representation theory and harmonic analysis of wreath products of finite groups, T. CECCHERINI-SILBERSTEIN, F. SCARABOTTI & F. TOLLI Moduli spaces, L. BRAMBILA-PAZ, O. GARCÍA-PRADA, P. NEWSTEAD & R.P. THOMAS (eds) Automorphisms and equivalence relations in topological dynamics, D.B. ELLIS & R. ELLIS Optimal transportation, Y. OLLIVIER, H. PAJOT & C. VILLANI (eds) Automorphic forms and Galois representations I, F. DIAMOND, P.L. KASSAEI & M. KIM (eds) Automorphic forms and Galois representations II, F. DIAMOND, P.L. KASSAEI & M. KIM (eds) Reversibility in dynamics and group theory, A.G. O’FARRELL & I. SHORT ˇ & M. POPA (eds) Recent advances in algebraic geometry, C.D. HACON, M. MUSTAT¸ A The Bloch–Kato conjecture for the Riemann zeta function, J. COATES, A. RAGHURAM, A. SAIKIA & R. SUJATHA (eds) The Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations, J.C. MEYER & D.J. NEEDHAM Arithmetic and geometry, L. DIEULEFAIT et al (eds) O-minimality and Diophantine geometry, G.O. JONES & A.J. WILKIE (eds) Groups St Andrews 2013, C.M. CAMPBELL et al (eds) Inequalities for graph eigenvalues, Z. STANIC´ Surveys in combinatorics 2015, A. CZUMAJ et al (eds) Geometry, topology and dynamics in negative curvature, C.S. ARAVINDA, F.T. FARRELL & J.-F. LAFONT (eds) Lectures on the theory of water waves, T. BRIDGES, M. GROVES & D. NICHOLLS (eds) Recent advances in Hodge theory, M. KERR & G. PEARLSTEIN (eds) Geometry in a Fréchet context, C.T.J. DODSON, G. GALANIS & E. VASSILIOU Sheaves and functions modulo p, L. TAELMAN Recent progress in the theory of the Euler and Navier–Stokes equations, J.C. ROBINSON, J.L. RODRIGO, W. SADOWSKI & A. VIDAL-LÓPEZ (eds) Harmonic and subharmonic function theory on the real hyperbolic ball, M. STOLL Topics in graph automorphisms and reconstruction (2nd Edition), J. LAURI & R. SCAPELLATO Regular and irregular holonomic D-modules, M. KASHIWARA & P. SCHAPIRA Analytic semigroups and semilinear initial boundary value problems (2nd Edition), K. TAIRA Graded rings and graded Grothendieck groups, R. HAZRAT Groups, graphs and random walks, T. CECCHERINI-SILBERSTEIN, M. SALVATORI & E. SAVA-HUSS (eds) Dynamics and analytic number theory, D. BADZIAHIN, A. GORODNIK & N. PEYERIMHOFF (eds) Random walks and heat kernels on graphs, M.T. BARLOW Evolution equations, K. AMMARI & S. GERBI (eds) Surveys in combinatorics 2017, A. CLAESSON et al (eds) Polynomials and the mod 2 Steenrod algebra I, G. WALKER & R.M.W. WOOD Polynomials and the mod 2 Steenrod algebra II, G. WALKER & R.M.W. WOOD Asymptotic analysis in general relativity, T. DAUDÉ, D. HÄFNER & J.-P. NICOLAS (eds) Geometric and cohomological group theory, P.H. KROPHOLLER, I.J. LEARY, C. MARTÍNEZ-PÉREZ & B.E.A. NUCINKIS (eds) Introduction to hidden semi-Markov models, J. VAN DER HOEK & R.J. ELLIOTT Advances in two-dimensional homotopy and combinatorial group theory, W. METZLER & S. ROSEBROCK (eds) New directions in locally compact groups, P.-E. CAPRACE & N. MONOD (eds) Synthetic differential topology, M.C. BUNGE, F. GAGO & A.M. SAN LUIS Permutation groups and cartesian decompositions, C.E. PRAEGER & C. SCHNEIDER Partial differential equations arising from physics and geometry, M. BEN AYED et al (eds) Topological methods in group theory, N. BROADDUS, M. DAVIS, J.-F. LAFONT & I. ORTIZ (eds) Partial differential equations in fluid mechanics, C.L. FEFFERMAN, J.C. ROBINSON & J.L. RODRIGO (eds) Stochastic stability of differential equations in abstract spaces, K. LIU Beyond hyperbolicity, M. HAGEN, R. WEBB & H. WILTON (eds)

London Mathematical Society Lecture Note Series: 455

Groups St Andrews 2017 in Birmingham

Edited by

C. M. CAMPBELL University of St Andrews C. W. PARKER University of Birmingham M. R. QUICK University of St Andrews E. F. ROBERTSON University of St Andrews C. M. RONEY-DOUGAL University of St Andrews

University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06-04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108728744 DOI : 10.1017/9781108692397 © Cambridge University Press 2019 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2019 Printed and bound in Great Britain by Clays Ltd, Elcograf S.p.A. A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Groups St. Andrews (Conference) (2017 : University of Birmingham) | Campbell, C. M., 1942- editor. Title: Groups St. Andrews 2017 in Birmingham / edited by C.M. Campbell [and four others]. Description: Cambridge ; New York, NY : Cambridge University Press, [2019] | Series: London Mathematical Society lecture note series ; 455 | Includes bibliographical references. Identifiers: LCCN 2018058651 | ISBN 9781108728744 (hardback : alk. paper) Subjects: LCSH: Group theory--Congresses. | Algebra--Congresses. Classification: LCC QA174 .G7744 2017 | DDC 512/.2--dc23 LC record available at https://lccn.loc.gov/2018058651 ISBN

978-1-108-72874-4 Paperback

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CONTENTS

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Finite simple groups and fusion systems Michael Aschbacher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Finite and infinite quotients of discrete and indiscrete groups Pierre-Emmanuel Caprace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Local-global conjectures and blocks of finite simple groups Radha Kessar & Gunter Malle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 A survey on some methods of generating finite simple groups Ayoub B. M. Basheer & Jamshid Moori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 One-relator groups: an overview Gilbert Baumslag, Benjamin Fine & Gerhard Rosenberger . . . . . . . . . . . . . . . . . . . . 119 New progress in products of conjugacy classes in finite groups Antonio Beltr´ an, Mar´ıa Jos´e Felipe & Carmen Melchor . . . . . . . . . . . . . . . . . . . . . . 158 Aspherical relative presentations all over again William A. Bogley, Martin Edjvet & Gerald Williams . . . . . . . . . . . . . . . . . . . . . . . . 169 Simple groups, generation and probabilistic methods Timothy C. Burness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 Irreducible subgroups of simple algebraic groups – a survey Timothy C. Burness & Donna M. Testerman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 Practical computation with linear groups over infinite domains A. S. Detinko & D. L. Flannery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 Beauville p-groups: a survey Ben Fairbairn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .271 Structural criteria in factorised groups via conjugacy class sizes Mar´ıa Jos´e Felipe, Ana Mart´ınez-Pastor & V´ıctor Manuel Ortiz-Sotomayor . . 289 Growth in linear algebraic groups and permutation groups: towards a unified perspective Harald A. Helfgott . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

vi L2 -Betti numbers and their analogues in positive characteristic Andrei Jaikin-Zapirain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 On the pronormality of subgroups of odd index in finite simple groups Anatoly S. Kondrat’ev, Natalia Maslova & Danila Revin . . . . . . . . . . . . . . . . . . . . . 406 Vertex stabilizers of graphs with primitive automorphism groups and a strong version of the Sims conjecture Anatoly S. Kondrat’ev & Vladimir I. Trofimov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 On the character degrees of a Sylow p-subgroup of a finite Chevalley group G(pf ) over a bad prime Tung Le, Kay Magaard & Alessandro Paolini . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 Patterns on symmetric Riemann surfaces Adnan Meleko˘glu & David Singerman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 Subgroups of twisted wreath products P´eter P. P´ alfy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 Some remarks on self-dual codes invariant under almost simple permutation groups B. G. Rodrigues & T. M. Mudziiri Shumba . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 Test elements: from pro-p to discrete groups Ilir Snopce & Slobodan Tanushevski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486

INTRODUCTION Groups St Andrews 2017 was held at the University of Birmingham from 5th August to 13th August 2017. This was the tenth in the series of Groups St Andrews group theory conferences. There were over 200 mathematicians involved in the meeting as well as some family members and partners. The Scientific Organising Committee of Groups St Andrews 2017 was Colin Campbell, Martyn Quick, Edmund Robertson and Colva Roney-Dougal (all from St Andrews) and Chris Parker (Birmingham). The academic business of the conference ran for seven days from Sunday 6th August to Saturday 12th August. Four main speakers delivered four talks each, surveying areas of contemporary development in group theory and related areas: Michael Aschbacher (Caltech), Pierre-Emmanuel Caprace (Universit´e Catholique de Louvain), Radha Kessar (City, University of London) and Gunter Malle (TU Kaiserslautern). There were five invited speakers delivering one-hour plenary talks: Tim Burness (University of Bristol), Vincent Guirardel (Universit´e de Rennes 1), Harald Helfgott (University of G¨ottingen), Andrei Jaikin-Zapirain (Universidad ´ Aut´ onoma de Madrid) and Donna Testerman (Ecole Polytechnique F´ed´erale de Lausanne). In addition there were about 115 contributed short talks from the delegates. In the evenings throughout the conference there was an extensive social programme. The main conference outing was a choice between Stratford upon Avon (with a chance to visit buildings and sites associated with William Shakespeare) and Warwick Castle. Other highlights of the social programme were a wine reception, a musical evening and the conference dinner. Once again T˙ Daily Group T˙ori< was a nice feature of the conference. We thank the various editors of this, by now traditional, publication. The support of the two main United Kingdom mathematics societies, the Edinburgh Mathematical Society and the London Mathematical Society has, once again, been an important factor in the success of these conferences. As well as supporting some of the expenses of the main speakers, the grants from these societies were used to support postgraduate students and also participants from Scheme 5 countries. Financial support was also received from the Heilbronn Institute for Mathematical Research and the publishers de Gruyter, Elsevier and Springer Nature. Once again all the main speakers have written substantial surveys, in one case a joint article, for these Proceedings. The other papers are also of a survey nature. We would like to thank Martyn Quick and Colva Roney-Dougal not only for their editorial assistance with these Proceedings but also for all their hard work in organising the conference. Last, and by no means least, we would like to thank Chris Parker, the local hero, for all his untiring efforts in making Birmingham such a happy venue. CMC, EFR

FINITE SIMPLE GROUPS AND FUSION SYSTEMS MICHAEL ASCHBACHER California Institute of Technology, Pasadena, California 91125, USA Email: [email protected]

This expository paper is taken from a series of four talks given at the conference Groups St Andrews in Birmingham 2017, held in August of 2017. The goal of those talks was to give the audience some insight into an ongoing program to, first, classify a certain class of simple 2-fusion systems, and then, second, to use the result on fusion systems to simplify the proof of the theorem classifying the finite simple groups (CFSG). But since the talks were delivered to a general audience of group theorists, most of the presentation was devoted to supplying background. The same is true of this article, where the program does not formally make an appearance until fairly late in the game. Thus we’ll begin with an introduction to the basic theory of fusion systems. Then we give an overview of the proof of that part of the CFSG devoted to the groups of component type, after which we discuss how to translate that proof into the category of 2-fusion systems, and indicate some advantages that accrue from that translation. We also describe some other changes to the original proof of the CFSG that are part of the program. Our basic reference on fusion systems is [2], although [7] also supplies a good introduction to the subject. Our basic reference on finite groups is [1]. For a more detailed discussion of the proof of the CFSG see [3].

Fusion systems Let p be a prime and S a finite p-group. A fusion system on S is a category F whose objects are the subgroups of S and, for subgroups P, Q of S, the set homF (P, Q) of morphisms from P to Q is a set of injective group homomorphisms of P into Q, and that set satisfies two weak axioms: (1) If s ∈ S with P s ≤ Q then the conjugation map cs : P → Q is a morphism. (2) If φ : P → Q is a morphism, then so is φ : P → P φ and φ−1 : P φ → P . Call S the Sylow group of F. Example 1.1 Let G be a finite group, S ∈ Sylp (G), and FS (G) the fusion system on S whose morphisms are induced via conjugation in G. Call FS (G) the p-fusion system of G. We are primarily interested in saturated fusion systems. A fusion system F is saturated if it satisfies two more axioms, that can be easily seen to hold in Example 1.1 using Sylow’s Theorem. See [2] for the axioms. This work was partially supported by DMS NSF-1265587 and DMS NSF-1601063.

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A saturated system F is exotic if it is the fusion system of no finite group; there exist exotic systems, and indeed for p odd they seem to proliferate. However 2-fusion systems seem to be more well behaved.

Luis Puig, modular representation theory, and algebraic topology The notion of a fusion system and much of the basic theory of fusion systems is due to Luis Puig, except that Puig uses different terminology and notation; see for example [13]. Puig’s primary interest is modular representation theory. I’m using notation and terminology due to some algebraic topologists, particular Broto, Levi, and Oliver in [6], since I learned the subject from their papers, and their terminology has by now become standard. In short, fusion systems originally arose in the context of modular representation theory, and remain of significant interest in that area. And of course algebraic topologists contribute to, and make use of, the theory of fusion systems; for example the Martino-Priddy Conjecture (now a theorem [11], [12]) says the p-completed classifying spaces of a pair of finite groups are of the same homotopy type precisely when their p-fusion systems are isomorphic. But I’m going to say no more about the role of fusion systems in representation theory and topology, and instead focus on the relationship between fusion systems and local finite group theory. However it should be noted that one of the advantages of the fusion system approach is that it draws on both topology and algebra.

A functor Let F be a fusion system on S. If F˜ is a fusion system on S˜ then a morphism from F to F˜ is a group homomorphism α : S → S˜ such that α induces a map from mor˜ For example if α = β|S for some homomorphism phisms of F to morphisms of F. ˜ is a morphism of fusion systems. ˜ then α : FS (G) → F ˜ (G) β:G→G S Indeed let G be the category whose objects are pairs (G, S) with G a finite ˜ S) ˜ a group group and S ∈ Sylp (G), and with a morphism from (G, S) to (G, ˜ with Sβ ≤ S. ˜ Then we have a functor (G, S) → FS (G) homomorphism β : G → G and β → β|S from G to the category of saturated fusion systems. The game is to use this functor to translate notions involving finite groups to analogous notions concerning fusion systems, and to prove theorems in one of the two categories using an analogous theorem in the other category.

A local theory of fusion systems For P ≤ S the set P F of conjugates of P consists of the images P φ, φ ∈ homF (P, S). In Example 1.1, P F is the set of G-conjugates of P contained in S. Define P to be fully normalized, fully centralized if for each Q ∈ P F , we have |NS (P )| ≥ |NS (Q)|, |CS (P )| ≥ |CS (Q)|, respectively. In Example 1.1, P is fully

Aschbacher: Finite simple groups and fusion systems

3

normalized precisely when NS (P ) ∈ Sylp (NG (P )). Write F f for the set of fully normalized subgroups of S. The local theory of finite groups studies finite groups G from the point of view of the local subgroups of G. Here a p-local subgroup of G is the normalizer NG (P ) of some nontrivial p-subgroup P of G. What is the right notion of a local subsystem of F? Let P ≤ S and define NF (P ), CF (P ) to be the subfusion system E of F with Sylow group T = NS (P ), CS (P ), such that for Q ≤ T , φ ∈ homF (Q, T ) is an E-morphism if and only if φ extends to ϕ ∈ homF (P Q, P T ) with P ϕ = P , ϕ centralizing P , respectively. In Example 1.1, if P is fully normalized then NF (P ) = FNS (P ) (NG (P )). The systems NF (P ) for 1 = P ∈ F f , play the role of the local subsystems of F. We want our local subsystems to be saturated; this follows from the following fundamental lemma of Puig: Lemma 1.2 (Puig) Let F be saturated and P ∈ F f . Then NF (P ) and CF (P ) are saturated. Define a subgroup P of S to be normal in F, and write P  F, if F = NF (P ). In Example 1.1 if P  G then P  F = FS (G), but the converse is not in general true. For example let G be a nonabelian finite simple group with abelian Sylow p-subgroup S and F = FS (G). Then by Burnside’s Fusion Theorem we have F = NF (S), so S  F but of course S is not normal in G. The product of normal subgroups of F is normal in F, so F has a largest normal subgroup, which we denote by Op (F). A subgroup P of S is centric if for each Q ∈ P F we have CS (Q) ≤ Q. In Example 1.1, P is centric if and only if P contains each p-element in CG (P ). Write F c for the set of centric subgroups of S. Next P is radical if Inn(P ) = Op (AutF (P )). Write F r for the set of radical subgroups and F f rc for the set of fully normalized, radical, centric subgroups. Remark 1.3 Let F be saturated. (1) One of the axioms of saturation says that if P ∈ F f then AutS (P ) ∈ Sylp (AutF (P )). (2) S ∈ F f rc . (3) If P ∈ F f rc − {S} then AutF (P ) is not a p-group. It is easy to see that S ∈ F f c , while S is radical by (1). If P ∈ F f rc and AutF (P ) is a p-group, then as P is radical we have Inn(P ) = AutF (P ). But by (1), AutS (P ) is Sylow in AutF (P ), so AutS (P ) = Inn(P ). Therefore NS (P ) = P CS (P ), and as P is centric we have CS (P ) ≤ P , so that NS (P ) = P , and hence S = P . That is (3) holds.

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Generation Let T ≤ S and Δ a set of morphisms between subgroups of T . The intersection of any collection of fusion systems on T is again a fusion system on T . Thus there is a smallest fusion system on T containing Δ, which we denote by Δ T . Call this system the subsystem on T generated by Δ. Theorem 1.4 (Alperin’s Fusion Theorem) Assume F is saturated. Then F = AutF (R) : R ∈ F f rc S . Remark 1.5 Observe that if Ξ is a set of representatives for the orbits of S on F f rc then F = AutF (R) : R ∈ Ξ S . Example 1.6 Let p = 2 and S = Dm be dihedral of order m > 4. Let’s determine, up to isomorphism, the saturated fusion systems on S. First S has two conjugacy classes EiS , i = 1, 2 of 4-subgroups, with E1 and E2 fused in Aut(S). Moreover for P ≤ S, Aut(P ) is not a 2-group if and only if P ∼ = E4 , in which case Aut(P ) = GL(P ) ∼ = S3 . It follows from Remark 1.3 that for R ≤ S we have R ∈ F f rc if and only if R = S or R is a 4-group with AutF (R) = Aut(R), and with R ∈ EiS for some i in the last case. Hence by Remark 1.5, up to isomorphism there are three potential saturated fusion systems on S: ∼ Z2 for i = 1, 2. (1) F0 where AutF (Ei ) = AutS (Ei ) = (2) F1 where AutF (E1 ) = Aut(E1 ) ∼ = S3 and AutF (E2 ) = AutS (E2 ) ∼ = Z2 . ∼ S3 for i = 1, 2. (3) F2 where AutF (Ei ) = Aut(Ei ) =

For 0 ≤ j ≤ 2 let Gj be a finite group with S ∈ Syl2 (Gj ) such that G0 = S, G1 ∼ = P GL2 (q1 ), and G2 ∼ = L2 (q2 ) for suitable odd qj . Then Fj = FS (Gj ), so Fj is saturated. Notice this proof also shows that there are exactly four saturated fusion systems on S, two of which are isomorphic via an outer automorphism of S. This is a toy example, but still it begins to suggest one approach to identifying a saturated fusion system F: find a small collection of “nice” subsystems of F that generate F, and show the corresponding amalgam of fusion systems is determined up to isomorphism by some suitable list of properties.

Factor systems Given a group G, the homomorphic images of G are the factor groups G/H for H  G, so such images are parameterized by the normal subgroups of G. The morphic images of a fusion system F are parameterized by the strongly closed subgroups of S. A subgroup T of S is strongly closed in S with respect to F if for each t ∈ T , we have tF ⊆ T . In Example 1.1, if H  G then S ∩ H is strongly closed in S with respect to FS (G).

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Let T be strongly closed in S. We can define a fusion system F/T on S/T such that the natural map s → sT is a surjective morphism of fusion systems from F onto F/T . The construction is the only one that could possibly work, and it is easy to show it works when T  F; in the general case, some effort is required. If F is saturated, then so is F/T . In Example 1.1 if H  G and T = S ∩ H then F/T ∼ = FSH/H (G/H). Later we will define the notion of a “normal subsystem” of a saturated fusion system. If E  F has Sylow group T then T is strongly closed and we can define the factor system F/E to be F/T .

Finite simple groups We now, for the moment, leave the topic of fusion systems, and consider instead the finite simple groups and their classification. Recall: Theorem 1.7 (Classification Theorem) Each finite simple group is isomorphic to one of the following: (1) A group of prime order. (2) An alternating group An , for some n ≥ 5. (3) A finite simple group of Lie type. (4) One of 26 sporadic simple groups. I’ll assume we are all familiar with the groups of prime order and the alternating groups. The groups of Lie type are linear groups, so each has an associated prime: the characteristic of the field of the defining vector space; call this prime the characteristic of the group. The sporadic groups live in a natural way in no known infinite family of simple groups. Eventually we will want to consider the 2-fusion systems of the simple groups, and use our functor to get information about those systems, and about simple 2-fusion systems in general. But first I want to discuss part of the proof of the Classification Theorem. To begin we need a few concepts and the associated notation.

The generalized Fitting subgroup Let G be a finite group. Define G to be quasisimple if G = [G, G] and G/Z(G) is simple. The components of G are its subnormal quasisimple subgroups, where subnormality is the transitive extension of the normality relation on subgroups of G. Let E(G) be the product of the components of G; it turns out that E(G) is a central product of the components: that is distinct components commute elementwise. Let F (G) be the largest normal nilpotent subgroup of G and F ∗ (G) = F (G)E(G); then F ∗ (G) is the central product of F (G) with E(G). We call F ∗ (G) the generalized Fitting subgroup of G.

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It turns out that CG (F ∗ (G)) = Z(F ∗ (G)), so F ∗ (G) controls the structure of G in the sense that the image of G in Aut(F ∗ (G)) under the conjugation map is isomorphic to G/Z(F ∗ (G)). Thus we can retrieve G, with little loss of information, from its generalized Fitting subgroup. See [1] for a detailed discussion of the generalized Fitting subgroup. The generalized Fitting subgroup is one of the important basic notions in the local theory of finite groups; this will become evident after more discussion. Define O(G) to be the largest normal subgroup of G of odd order; Gorenstein called O(G) the core of G. The CFSG focuses on 2-local subgroups of G; the cores of 2-locals cause significant difficulties in the CFSG. One of the advantages of working with 2-fusion systems is that such difficulties vanish, since cores disappear when we apply our functor, as the following lemma suggests: Lemma 1.8 Let ∗ : G → G/O(G) = G∗ be the natural homomorphism ∗ : g → gO(G) = g ∗ and S ∈ Syl2 (G). Then ∗ : FS (G) → FS ∗ (G∗ ) is an isomorphism. One consequence of Lemma 1.8 is that if F is the 2-fusion system of a finite group then F is the 2-fusion system of an infinite number of finite groups. Hence it would seem that when applying our functor from finite groups to fusion systems, we lose a lot of information. While this is true, it may also be true that the lost information only serves to confuse many issues, and it may be an advantage to discard it. Let L0 be the preimage in G of E(G/O(G)) and L(G) = L∞ 0 be the last term in the derived series for L0 . We call L(G) the layer of G. Observe that L0 = L(G)O(G). The following result is due to Gorenstein and Walter; see [1] for a proof, modulo an appeal to the Schreier Conjecture. Theorem 1.9 (L-Balance Theorem) L(CG (P )) ≤ L(G).

For each 2-subgroup P of G we have

The groups of Lie type of characteristic 2 have a different 2-local structure than those of odd characteristic. We seek to capture that difference in the general finite group in abstract group theoretic terms, rather than in the context of linear groups. Define G to be of component type if L(CG (t)) = 1 for some involution t of G; roughly speaking, the centralizer in G of some involution has a component. Define G to be of characteristic 2-type if F ∗ (H) = O2 (H) for each 2-local subgroup H of G. Remark 1.10 If G is a simple group of Lie type and even characteristic, then G is of characteristic 2-type. On the other hand almost all simple groups of Lie type and odd characteristic, other than L2 (q), are of component type. The alternating groups An for n > 8 are of component type. Some sporadic groups are of component type and some are of characteristic 2-type. In short, if we seek to partition the simple groups into “even” and “odd” groups in terms of their 2-local structure, and in such a way that the groups of Lie type

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and even characteristic are “even”, while those of Lie type and odd characteristic are odd, then we are led to define the even groups to be those of characteristic 2-type and the odd groups to be those of component type. Later we will see that this odd-even partition of the simple groups is perhaps not the optimal choice. But first let us see that, at least generically, each simple group is either odd or even using this definition: Theorem 1.11 (Gorenstein-Walter Dichotomy Theorem) Assume O(G) = 1 and m2 (G) > 2. Then G is of component type or characteristic 2-type. See [3] for a proof of the Dichotomy Theorem. Here m2 (G) is the 2-rank of G: the maximum m such that G contains a subgroup that is the direct product of m groups of order 2. The groups of 2-rank 2 should be thought of as “small” groups. Thus the Dichotomy Theorem says that, generically, each core-free finite group is either odd or even. Then the proof of the CFSG treats the small simple groups, the odd simple groups, and the even simple groups, using different methods for each type of group. We are interested in simplifying the treatment of the odd simple groups. The most obvious advantage gained by treating the odd simple groups (as odd fusion systems) in the category of 2-fusion systems, comes from avoiding obstructions presented by cores of 2-locals, since, by Lemma 1.8, these cores vanish when we apply our functor. In the treatment of groups of component type, the biggest obstacle mounted by core obstruction arises from the necessity to verify the B-Conjecture: B-Conjecture. E(CG (t)).

If O(G) = 1 then for each involution t in G, we have L(CG (t)) =

The proof of the B-Conjecture is difficult and indirect. See [3] for more discussion of the B-Conjecture. Given a simple group G of component type, and assuming the B-Conjecture, we can consider the set C(G) of components of centralizers of involutions. If L is a component of CG (t) for some involution t, s is an involution centralizing t and L, and Gs = CG (s), then L is a component of CGs (t), so by L-Balance and the B-Conjecture, we have L ≤ L(Gs ) = E(Gs ). Indeed there exists a component K of Gs such that either K = K t and L = E(CKK t (t)) is an image of K, or L pumps up to K: K = K t and L is a component of CK (t). Keeping track of the pump up “ordering” on C(G) and playing some combinatorial games, we are able to pin down the centralizer of an involution possessing a “maximal” member of C(G). Then, as in the Brauer program, we identify G from this centralizer. I’ll be a bit more precise about what such a centralizer looks like later. We seek to make an analogous argument in the category of 2-fusion systems. To do so, we must translate notions like “simple”, “quasisimple”, “component”, etc., and theorems like L-Balance and the Dichotomy Theorem into analogous results on 2-fusion systems. The first crucial step in that process is to identify a notion of “normal subsystem” of a saturated fusion system.

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Normal subsystems Let F be a fusion system on a p-group S. We begin with the notion of an Finvariant subsystem. There are at least three equivalent definitions of such a system; here is one. Let E be a subsystem of F on T . We say that E is F-invariant if T is strongly closed in S with respect to F and for each P ≤ Q ≤ T , each φ ∈ homE (P, Q), and each α ∈ homF (Q, S), we have φα∗ ∈ homE (P α, T ), where φα∗ = α−1 φα. The notion of F-invariance is well behaved, but it has one draw back: even when F is saturated, an invariant subsystem need not be saturated. Fortunately there is an easy way to correct this. Assume F is saturated and define a subsystem E of F to be weakly normal in F if E is F-invariant and saturated. Finally E is normal in F if E is weakly normal in F and satisfies the extension condition: for each α ∈ AutE (T ), α extends to α ˆ ∈ AutF (T CS (T )) such that [α ˆ , CS (T )] ≤ Z(T ). Write E  F to indicate that E is normal in F. If P is a subgroup of S normal in F then FP (P )  F. In Example 1.1, if H  G then FS∩H (H)  FS (G). The converse is in general false; as we saw in an earlier example, if G is simple and S abelian then S  FS (G) but S is not normal in G. Define F to be constrained if there is a centric subgroup of F normal in F. In Example 1.1, if F ∗ (G) = Op (G) then FS (G) is constrained as CG (F ∗ (G)) = Z(F ∗ (G)). Define a model of a constrained system F to be a group G with F ∗ (G) = Op (G) and FS (G) = F. The topologists have shown in [5] that: Theorem 1.12 (Model Theorem) If F is a constrained saturated fusion system then F has a model G, and G is unique up to an isomorphism which is the identity on S. Theorem 1.13 Let F be a constrained saturated fusion system with model G. Then the map H → FS∩H (H) is a bijection between the normal subgroups of G and the normal subsystems of F. The invariance condition is part of the definition of “normal subsystem” to insure our functor is bijective in Theorem 1.13. Given a notion of “normal subsystem”, we can now translate many notions from finite group theory to analogous notions about saturated fusion systems. As in the case of groups, subnormality for fusion systems is the transitive extension of the normality relation. Our saturated system F is simple if it has no nontrivial normal subsystem. There is a smallest normal subsystem E of F such that F/E is the system of a p-group; denote this system by Op (F). Define F to be quasisimple if F = Op (F) and F/Z(F) is simple. Define the components of F to be its subnormal quasisimple subsystems. It can be shown that F has a normal subsystem E(F) that is the central product of the components of F. Further E(F) centralizes Op (F) and F has a normal

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subsystem F ∗ (F) which is a central product of Op (F) and E(F). Call F ∗ (F) the generalized Fitting subsystem of F; it can be shown that CF (F ∗ (F)) = Z(F ∗ (F)). Theorem 1.14 (E-Balance Theorem) For each P ∈ F f , E(CF (P )) ≤ E(F). Define F to be of characteristic p-type if for each 1 = P ∈ F f , we have NF (P ) is constrained. Define F to be of component type if for some P ∈ F f of order p, E(CF (P )) = 1. Theorem 1.15 (Dichotomy Theorem for Fusion Systems) Let F be a saturated fusion system on a p-group S. Then F is either of characteristic p-type or of component type. The Dichotomy Theorem for fusion systems is stronger and has a more elegant statement than the Dichotomy Theorem for groups. It is also easier to prove.

Beginning the program Given the Dichotomy Theorem for Fusion Systems it makes sense to attempt to classify the simple 2-fusion systems of component type using the classification of the simple groups of component type as a template. In actual fact I propose to do something a bit different, but a discussion of those changes is perhaps best put off for a while. The CFSG proceeds by induction on the group order, so one considers a simple group of minimal order subject to not being on the list of “known” simple groups. In such a group G each proper simple section of G is known. We will make a related assumption on our fusion systems. ˜ the class of Let K be the class of “known” simple 2-fusion systems, and K “known” quasimple 2-fusion systems: those whose central factor system is in K. I’ll say a few words about these two classes shortly. Let F be a saturated fusion system on a 2-group S. Define C(F) to be the set of components of centralizers of involutions in F; that is C ∈ C(F) if there exists ¯ of (t, C) such that t¯ is fully centralized some involution t in S and a conjugate (t¯, C) ¯ ¯ and C is a component of CF (t). Thus F is of component type if C(F) is nonempty. ˜ We will assume that each member of C(F) is in K. ¯ ¯ Notice that we must pass to a conjugate (t, C) with t¯ fully centralized, so that we can apply Lemma 1.2 to insure that CF (t¯) is saturated. This is necessary as components are only defined for saturated systems. As in the CFSG, we have the pump up relation on C(F), and we wish to show that if C is “maximal” with respect to this relation then the centralizers of involutions centralizing C are controlled. Finally we want to show the existence of such centralizers forces F to be isomorphic to a member of K. Let us see in more detail what this means for groups: Let G be a finite group with O(G) = 1, satisfying the B-conjecture. Let L ∈ C(G) have no proper pumpups, and set K = CG (L). Then (essentially) either (1) L ∈ Comp(G), or

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(2) L is standard in G: that is NG (L) = NG (K), L commutes with none of its conjugates, and K is tightly embedded in G. Here a subgroup K of G is tightly embedded in G if |K| is even, but |K ∩ K g | is odd for all K = K g . It can be shown that in case (2), a Sylow 2-subgroup Q of K is small; namely either m2 (K) = 1 or Q is elementary abelian, and then in the latter case, even Q ∼ = E4 . Thus the centralizer CG (t) of an involution t in K closely resembles the centralizer CG¯ (t¯) of some involution t¯ in some known simple ¯ and this can be used to show G ∼ ¯ is known. group G, =G ∼ ¯ ¯∼ ¯ For example if G = An with n > 8 then G has a standard subgroup L = An−4 ¯ ∼ ¯ with Q . And if G is of Lie type over a field of odd order q, then usually E = 4 ¯ ∼ ¯ is ¯ has a standard subgroup L ¯ of Lie type over Fq with K G = SL2 (q), so that Q quaternion. As a first step toward proving K is tightly embedded in G, one uses the condition that L has no proper pumpups to show L ∈ Comp(CG (i)) for each involution i ∈ K. With a little care, it is possible to establish an analogous statement for fusion systems. One can also define the notion of a “tightly embedded subsystem” of a saturated fusion system, and prove the necessary theorems for such subsystems. But then we encounter a difficulty: Problem. If F is a saturated fusion system and E is a subsystem of F, we do not know how to define the normalizer or centralizer in F of E, except in very special situations. Because of the Problem, it is not straightforward to define a notion of a “standard subsystem” of a fusion system analogous to the notion of a standard subgroup defined above, but it is possible. In short, the necessary notions from the CFSG do not all translate to fusion systems in a straightforward manner, but by and large it seems that such difficulties can be overcome. We will return to such details later; first let us discuss K.

The class K of known simple 2-fusion systems Let F be a saturated fusion system on a 2-group S. Recall that F is exotic if F is realized by no finite group. There is one known class of exotic simple 2fusion systems: the exotic Benson-Solomon systems FSol (q), for q an odd prime power. If F is such a system then F has one class of involutions z F and CF (z) is the 2-fusion system of Spin7 (q), which is quasisimple, so F is of component type. The isomorphism type depends only on the 2-share (q 2 − 1)2 of q 2 − 1, not on q. The systems were “discovered” by Benson in a topological context, and earlier by Solomon as part of the CFSG, but these “discoveries” took place before the notion of a fusion system really existed. So assume F is simple and realized by a finite group G. Then it is easy to see that we may choose G to be simple, so we need to examine the known simple groups G to see when FS (G) is simple. A sufficient condition is to show that, first, S is the smallest nontrivial strongly closed subgroup of S, and, second, that

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a certain condition on AutF (S) is satisfied; for example it suffices to show that AutF (S) = Inn(S). Richard Foote [8] has determined those known simple groups G with a strongly closed subgroup proper in S: He shows such a G is either a group of Lie type in characteristic 2 of Lie rank 1, or S is abelian; call such a simple group a Goldschmidt group since, as part of the CFSG, Goldschmidt [9] proved that these are the simple groups with a strongly closed abelian 2-subgroup. In particular in such a G, S  FS (G), so FS (G) is not simple. In the remaining groups the condition on AutF (S), can be checked, so: Theorem 1.16 K consists of the exotic Benson-Solomon systems, together with the 2-fusion systems of the known simple groups that are not Goldschmidt groups.

Tame realization We’ve determined K, but we need to do more: we need to determine certain ex˜ so we must tensions of members D of K. In particular we need to describe K, determine the quasisimple extensions of D. Define F to be a small extension of D if E(F) is quasisimple with E(F)/Z(E(F)) ∼ = D. We want to determine the small extensions of the members of K. If D is a Benson-Solomon system then suitable small extensions have been described by ad hoc methods. The following result is due to Bob Oliver and various collaborators; some of the work has not yet appeared, but see [4] for a more detailed discussion and those references that have appeared. Theorem 1.17 Assume D is a simple system realized by some known finite simple group. Then (1) D is tamely realized by some known finite simple group K. (2) If F is a small extension of D then F is realized by some finite group G with F ∗ (G) = O2 (G)E(G) and E(G)/Z(E(G)) ∼ = K. In short, Theorem 1.17 reduces the description of suitable extensions of D to ˜ a description of certain group extensions of K. In particular the members of K are the Benson-Solomon systems together with the 2-fusion systems of the known quasisimple groups that are not Goldschmidt. The notion of tame realization depends upon the linking system of D, a topological notion I’ll avoid in these talks. Lemma 1.18 (Pump Up Lemma) Let t be a fully centralized involution in a saturated 2-fusion system F and C a component of CF (t). Then there exists a unique component L of F such that either (1) L = Lt and C = E(CLLt (t)) is diagonally embedded in the central product LLt , so D is a morphic image of L, or (2) C pumps up to L: that is L = Lt and C is a component of CL (t).

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If C ∈ C(F) is a component of the fully centralized involution t, and s is an involution centralizing C and fully centralized in CF (t), then there exists a conjugate ¯ of (t, s, C) with s¯ ∈ F f , t¯ ∈ Fs¯f , where Fs¯ = CF (¯ (t¯, s¯, C) s), and C¯ is a component ¯ of CFs¯ (t). Thus we can apply the Pump Up Lemma to Fs¯ in the role of F to conclude there exist L ∈ C(F) a component of Fs¯ such that either C¯ is diagonally embedded in LLt¯, or C¯ pumps up to L. In the latter case we say that C pumps up to L. Recall we focus on “maximal” members of C(F): those with no proper pump ups. Unfortunately there exist examples of groups with an involution t such that CG (t) = t × K for some simple group K and K does not pump up in C(G), but K is so small (e.g., |G| > |K|2 , or perhaps more to the point |G|2 > |K|22 ) that it is difficult to identify G from CG (t). Such examples are not simple but it takes a lot of effort to see this. For example take G to be the wreath product of K by Z2 , where K is diagonally embedded in E(G) ∼ = K × K. Or take L to be a group X(q 2 ) of Lie type over Fq2 with q even and G to be the extension of L by an involutory field automorphism t; thus CL (t) has a component isomorphic to X(q). Both these examples possess a standard subgroup as defined earlier, but in these cases the standard subgroup is a poor vehicle for identifying G, so we should adjust our approach so as to avoid them. Notice in particular that neither group G is simple. Such examples have analogues in the fusion system universe, which we wish to avoid. The GLS revision program reorganizes the original proof of the CFSG, in part to avoid many such examples. I will describe a different approach.

Intrinsic members of C(F ) Let F be a saturated fusion system on a 2-group S. Define C ∈ C(F) to be intrinsic ¯ in C(F) if for some conjugate C¯ of C there is a fully centralized involution t¯ ∈ Z(C) such that C¯ is a component of CF (t¯). Define C to be subintrinsic in C(F) if there ¯ intrinsic in C(C) ¯ such that D is also intrinsic in C(F). Subintrinsic is D ∈ C(C) components do not have the bad behavior discussed above. Here is an important result on intrinsic components; it is the fusion theoretic version of the Classical Involution Theorem for groups. Theorem 1.19 Let F be a saturated 2-fusion system such that for some fully centralized involution z there is a subnormal subsystem E of CF (z) such that z ∈ E and E is the 2-fusion system of SL2 (q) for some odd q. Assume F is the normal closure of a Sylow group of E. Then F is the 2-fusion system of some quasisimple group of Lie type and odd characteristic. Theorem 1.19 can be used to show that if F is simple and C(F) contains the 2-fusion system of a quasisimple group of Lie type and odd characteristic, other than L2 (q), then almost always F ∈ K.

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J-components Let J be the set of involutions j in S such that m2 (CS (j)) = m2 (S). Define CJ (F) ¯ of to consist of those C ∈ C(F) for which there exists t ∈ J and a conjugate (t¯, C) ¯ ¯ ¯ (t, C) such that t is fully centralized and C is a component of CF (t). Members of CJ (F) do not have the bad behavior discussed above. Define an almost simple 2-fusion system F with F = O2 (F) to be odd if either (a) there exists C subintrinsic in C(F), or (b) CJ (F) is nonempty. Define F to be even if F is not odd. Therefore F is even if either F is of characteristic 2-type, or F is of component type but not odd. Remark 1.20 The Benson-Solomon systems FSol(q) are odd, since C(FSol(q) ) possesses an intrinsic member which is the 2-fusion system of Spin7 (q). Suppose F = FS (K) for some known simple group K. If K is of Lie type and characteristic 2, then F is of characteristic 2-type, and hence even. If K is of Lie type and odd characteristic, but not L2 (q), then almost always F is odd. If K ∼ = An for n ≥ 10 is alternating, then F is odd. Behavior when K is sporadic is mixed. The first goal of our program is to solve the following problem: Odd 2-Fusion System Problem Assume F is an odd 2-fusion system such ˜ Prove E(F) is in K. that each member of C(F) is in K. The second goal is to define a corresponding class of “odd simple groups” whose 2-fusion systems are odd, and to use the theorem on odd fusion systems to show each odd simple group is known.

Odd simple groups Let G be a finite group. A 2-component of G is a subnormal subgroup H of G such that H = H ∞ and H/O(H) is quasisimple. Write L(G) for the set of 2-components of centralizers of involutions of G and L+ (G) for the set of images L/O(L) for L ∈ L(G). For example if O(G) = 1 and G satisfies the B-Conjecture then L(G) = C(G). For L a quasisimple group define L to be intrinsic if Z(L) is of even order. Write J for the set of involutions j ∈ G with m2 (CG (j)) = m2 (G). We will assume each member of L+ (G) is a known quasisimple group; if F is the ˜ 2-fusion system of G this assumption implies each member of C(F) is in K. Define a simple group G to be odd if either (a) L+ (G) contains an intrinsic member that is not Goldschmidt, or (b) for some j ∈ J and some 2-component L of CG (j), L/O(L) is not Goldschmidt.

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Theorem 1.21 Assume G is an almost simple group with G = O2 (G) such that each member of L+ (G) is known. Then the 2-fusion system F of G is almost simple and F = O2 (F). Odd Group Problem. Assume G is an odd simple group in which each member of L+ (G) is known. Prove that G is known. Theorem 1.21 says that if G satisfies the hypothesis of the Odd Group Problem, then its 2-fusion system F satisfies the hypothesis of the Odd 2-Fusion System Problem. Hence if the latter problem has a positive solution then E(F) ∈ K, and we should be able to use this fact to show that G is known, hence showing that the Odd Group Problem has a positive solution. For example assume that F = FS (K) for some group K of Lie type over Fq for q odd, other than L2 (q). Then, as in Theorem 1.19, there is a fully centralized involution z in S and a subnormal subsystem E of CF (z) with z ∈ E and E the 2-fusion system of SL2 (q). From the classification of groups with quaternion Sylow 2-subgroups, there is a subnormal subgroup L of CG (z) with 2-fusion system E such that L/O(L) ∼ = SL2 (q0 ) for some q0 . (We’ll ignore the case L/O(L) ∼ = Aˆ7 .) At this point we could appeal to the Classical Involution Theorem to conclude G is of Lie type in odd characteristic or M11 . But since we want to derive the Classical Involution Theorem from Theorem 1.19, we must take another approach. Roughly speaking, if K is of Lie rank at least 3, we can use signalizer functor theory to show that O(L) = 1, and then use the Curtis-Tits-Phan Theorem (cf. [10]), applied to a suitable set of conjugates of L, to identify G as a group of Lie type over Fq0 with 2-fusion system F. If the Lie rank of K is at most 2 some other argument is necessary. For example if K is SL2 (q) we need the classification of groups with quaternion Sylow 2-subgroups. If K is L3 (q) or U3 (q), we need the classification of groups with semidihedral or wreathed Sylow 2-subgroups. For larger rank 2 groups such as G2 (q) or 3 D4 (q), the fusion system F supplies more information about G, but more work is still necessary.

Even groups and 2-fusion systems It would be nice to classify the simple 2-fusion systems. If the Odd 2-Fusion System Problem has a positive solution, we would be reduced to even systems. In the CFSG, the groups G of characteristic 2-type with e(G) ≥ 3 were treated by passing to the p-local structure of G for suitable odd primes p. For obvious reasons, such an approach is not available for 2-fusion systems. However there is a program of Meierfrankenfeld, Stellmacher, and Stroth (MSS), aimed at determining groups of characteristic 2-type from the point of view of 2-locals. Perhaps methods can be borrowed from the MSS program to treat even 2-fusion systems. Alternatively, if one is only interested in the CFSG, then one can focus on the even groups: the simple groups G whose 2-fusion system is even. It turns out that this class of groups is closely related to the GLS class of groups of even type. So

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perhaps it is possible to hook up with the GLS program to determine the even simple groups. Finally MSS also work with a larger class of groups they call groups of parabolic characteristic 2. This class of groups includes our even groups, so if MSS can classify all such groups then we would have an alternate proof of the CFSG. References [1] M. Aschbacher, Finite Group Theory, Cambridge Univ. Press, 1986. [2] M. Aschbacher, R. Kessar, and B. Oliver, Fusion Systems in Algebra and Topology, Cambridge University Press, 2011. [3] M. Aschbacher, R. Lyons, S. Smith, and R. Solomon, The Classification of Finite Simple Groups; Groups of Characteristic 2 Type, Mathematical Surveys and Monographs, vol. 172, AMS, 2011. [4] M. Aschbacher and B. Oliver, Fusion systems, Bull. Amer. Math. Soc. 53 (2016), 555–615. [5] C. Broto, N. Castellana, J. Grodal, R. Levi, and B. Oliver, Subgroup families controlling p-local finite groups, Proc. London Math. Soc. 91 (2005), 325–354. [6] C. Broto, R. Levi, and B. Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc. 16 (2003), 779–856. [7] D. Craven, The Theory of Fusion Systems: An Algebraic Approach, Cambridge Univ. Press, 2011. [8] R. Foote, A characterization of finite groups containing a strongly closed 2-subgroup, Comm. Alg. 25 (1997), 593–606. [9] D. Goldschmidt, 2-Fusion in finite groups, Annals Math. 99 (1974), 70–117. [10] D. Gorenstein, R. Lyons, and R. Solomon, The Classification of the Finite Simple Groups, Number 3, Mathematical Surveys and Monographs, Vol. 40, Amer. Math. Soc., 1998. [11] B. Oliver, Equivalence of classifying spaces completed at odd primes, Math. Proc. Camb. Phil. Soc. 137 (2004), 321–347. [12] B. Oliver, Equivalence of classifying spaces completed at the prime two, Memoirs Amer. Math. Soc. 848 (2006). [13] L. Puig, Frobenius Categories versus Brauer Blocks, Birkhauser, 2009.

FINITE AND INFINITE QUOTIENTS OF DISCRETE AND INDISCRETE GROUPS PIERRE-EMMANUEL CAPRACE Universit´e catholique de Louvain, IRMP, Chemin du Cyclotron 2, bte L7.01.02, 1348 Louvain-la-Neuve, Belgique Email: [email protected]

Abstract These notes are devoted to lattices in products of trees and related topics. They provide an introduction to the construction, by M. Burger and S. Mozes, of examples of such lattices that are simple as abstract groups. Two features of that construction are emphasized: the relevance of non-discrete locally compact groups, and the two-step strategy in the proof of simplicity, addressing separately, and with completely different methods, the existence of finite and infinite quotients. A brief history of the quest for finitely generated and finitely presented infinite simple groups is also sketched. A comparison with Margulis’ proof of Kneser’s simplicity conjecture is discussed, and the relevance of the Classification of the Finite Simple Groups is pointed out. A final chapter is devoted to finite and infinite quotients of hyperbolic groups and their relation to the asymptotic properties of the finite simple groups. Numerous open problems are discussed along the way.

Contents 1 Just-infiniteness versus SQ-universality

17

2 Finitely generated infinite simple groups: historical landmarks 2.1 The first existence proof, after G. Higman . . . . . . . . . . . . . . . 2.2 The first explicit family, after R. Camm . . . . . . . . . . . . . . . . 2.3 The first finitely presented infinite simple group, after R. Thompson 2.4 Quotients of free amalgamated products of free groups . . . . . . . .

19 19 20 20 22

3 Kneser’s simplicity conjecture 23 3.1 The multiplicative group of the Hamiltonian quaternions . . . . . . . 23 3.2 The Margulis Normal Subgroup Theorem . . . . . . . . . . . . . . . 25 3.3 Finite quotients of the multiplicative group of a division algebra . . 26 4 Lattices in products of trees, after M. Burger, S. Mozes and D. Wise 27 4.1 BMW-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.2 Examples of BMW-groups of small degree . . . . . . . . . . . . . . . 30 P.-E.C. is a F.R.S.-FNRS senior research associate, supported in part by EPSRC grant no. EP/K032208/1.

Caprace: Finite and infinite quotients of discrete and indiscrete groups 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12

Inseparability and irreducibility . . . . . . . . . . . . . . . . . . Anti-tori and irreducibility . . . . . . . . . . . . . . . . . . . . . Local actions and irreducibility . . . . . . . . . . . . . . . . . . Residual finiteness . . . . . . . . . . . . . . . . . . . . . . . . . The Normal Subgroup Theorem, after U. Bader and Y. Shalom Alternating and fully symmetric local actions . . . . . . . . . . Virtually simple BMW-groups of small degree . . . . . . . . . . The hyperbolic manifold analogy . . . . . . . . . . . . . . . . . Local actions of just-infinite groups acting on trees . . . . . . . Lattices in products of more than two trees . . . . . . . . . . .

17 . . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

5 Quotients of hyperbolic groups and asymptotic properties of finite simple groups 5.1 Examples of hyperbolic groups . . . . . . . . . . . . . . . . . . . . . 5.2 Finite and infinite quotients of hyperbolic groups . . . . . . . . . . . 5.3 Hyperbolic quotients of hyperbolic groups, after A. Olshanskii . . . . 5.4 The space of marked groups . . . . . . . . . . . . . . . . . . . . . . . 5.5 Examples of fully residually finite simple groups . . . . . . . . . . . . 5.6 Virtual specialties, after I. Agol, F. Haglund and D. Wise . . . . . .

1

33 34 35 38 41 43 45 46 49 50

52 52 52 54 57 59 60

Just-infiniteness versus SQ-universality The true infinite is both finite and infinite: it is the overcoming of both infiniteness and finiteness. It is therefore pure indeterminacy and pure freedom. Carlos Alberto Blanco, Philosophy and salvation, 2012

The general theme of these notes is the normal subgroup structure of infinite groups. Two opposite extreme behaviors can be observed: groups with ‘few’ normal subgroups, like the simple groups, on one hand, and groups with ‘many’ normal subgroups, like the free groups, on the other hand. The following concepts provide a possible formal way to isolate two precise classes of groups corresponding to those two extremes. A group is called just-infinite if it is infinite and all its proper quotients are finite. In other words, a just-infinite group is an infinite group all of whose non-trivial normal subgroups are of finite index. Infinite simple groups are obvious examples; other basic examples are the infinite cyclic group and the infinite dihedral group. At the other extreme, a group G is called SQ-universal if every countable group embeds as a subgroup in some quotient of G. That term was coined by P. Neumann [91] following a suggestion of G. Higman. An emblematic result, due to Higman–Neumann–Neumann [62], is that the free group of rank 2 is SQ-universal. It is important to observe that these two properties are mutually exclusive for a countable group. Indeed, if a just-infinite countable group were SQ-universal, then it would contain an isomorphic copy of every finitely generated infinite group. This is impossible, since a countable group contains countably many finite subsets,

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while the 2-generated groups fall into uncountably many isomorphism classes by a classical result of B. H. Neumann [89, Theorem 14]. Just-infinite groups arise naturally in view of the following basic consequence of Zorn’s lemma: every finitely generated infinite group has a just-infinite quotient. An important theorem describing the structure of just-infinite groups was established by J. Wilson in his thesis, see [126], [127] and [53]. His results highlight the importance of the subclass consisting of the so-called hereditarily just-infinite groups, i.e., the infinite groups all of whose finite index subgroups are just-infinite. The wreath product S  C2 of an infinite simple group S with a cyclic group of order 2 is an example of a just-infinite group which is not hereditarily so. Let us record the following elementary fact. Proposition 1.1 Let G be a hereditarily just-infinite group. Either G is residually finite, or the intersection G(∞) of all its subgroups of finite index is simple and of finite index in G. Proof Suppose that G is not residually finite. Then G(∞) is a non-trivial normal subgroup of G. Hence [G : G(∞) ] is finite, so that G(∞) is itself just-infinite. Any finite index subgroup of G(∞) has finite index in G, and thus contains G(∞) by the definition of G(∞) . Therefore, the just-infinite group G(∞) has no proper subgroup of finite index. Hence, it is a simple group.  The intersection G(∞) of all finite index subgroups of G is called the finite residual of G. Proposition 1.1 suggests a two-step approach to show that a group G is virtually simple: one step is to show that G is hereditarily just-infinite, the other is that G is not residually finite. As we shall see in Section 3, it is precisely this approach that G. Margulis [84] used in his proof of Kneser’s conjecture on the simplicity of certain anisotropic simple algebraic groups over number fields. Moreover, nondiscrete locally compact groups are used in an essential way to establish the justinfiniteness, via Margulis’ Normal Subgroup Theorem (see Section 3.2), while the Classification of the Finite Simple Groups has been used extensively in more recent investigations of the normal subgroup structure of anisotropic simple algebraic groups (see Section 3.3). It is the same two-step strategy that M. Burger and S. Mozes [22] implemented in their celebrated construction of simple lattices in the automorphism group of a product of two regular locally finite trees. We give an overview of their work in Section 4, and present also the seminal ideas developed independently around the same time by D. Wise [130] concerning lattices in products of trees. In particular, we give an explicit presentation of a simple group, recently found by N. Radu [99], that splits as an amalgamated free product of two free groups of rank 3 over a common subgroup of index 5, see Section 4.9. A key idea due to Burger–Mozes, is to exploit some geometric aspects of finite group theory, which become relevant through the concept of the local action of a group of automorphisms of a connected locally finite graph, see Section 4.5.

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19

The contrast between the study of finite and infinite quotients of a finitely presented group is further illustrated in the final section, devoted to Gromov hyperbolic groups. While those are known to be either SQ-universal or virtually cyclic in view of a result independently established by T. Deltzant [44] and A. Olshanskii [94], it is a major open problem to determine whether they are all residually finite. We discuss that problem, and its relation to the asymptotic properties of the finite simple groups, notably by the consideration of the space of marked groups.

2 2.1

Finitely generated infinite simple groups: historical landmarks The first existence proof, after G. Higman

The question of existence of a finitely generated infinite simple group was asked by A. Kurosh [75] in 1944. A positive answer was given by G. Higman [60] a few years later. It is based on the following observation. Lemma 2.1 (G. Higman [60]) vides 2n − 1 is n = 1.

(i) The only positive integer n such that n di-

(ii) In any group G, the only element g of finite order that is conjugate to its square by an element of the same order as g is the trivial one. Proof (i). Suppose for a contradiction that n > 1 divides 2n − 1 and let p be the smallest prime divisor of n. Then p divides 2n − 1. Hence p is odd and the order of 2 in the multiplicative group F∗p is a divisor of n, as well as a divisor of p − 1. By the choice of p, the integer n is relatively prime to p − 1, and we get a contradiction. (ii). Let x ∈ G be an element of the same order as g ∈ G, say n, such that n n xgx−1 = g 2 . Then g = xn gx−n = g 2 , so that g 2 −1 = 1. The conclusion follows from (i).  Theorem 2.2 (G. Higman [60]) The group 2 H = a0 , a1 , a2 , a3 | ai ai+1 a−1 i = ai+1 , i mod 4

is infinite and its only finite quotient is the trivial one. In particular H has a finitely generated infinite simple quotient. Proof To see that H is infinite, one observes that it can be decomposed as a non-trivial amalgamated free product whose factors are themselves non-trivial amalgamated free products of proper HNN-extensions. In order to elucidate the finite quotients of H, let us observe that H has an automorphism of order 4 that cyclically permutes the generators. Therefore, if H has a non-trivial finite quotient, then H also has a non-trivial finite quotient in which the order of the image of ai is the same integer n for all i. Since ai conjugates ai+1 on its square, we deduce from Lemma 2.1 that the image of ai in the finite

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Caprace: Finite and infinite quotients of discrete and indiscrete groups

quotient in question is trivial for all i, which is absurd. This shows that the only finite quotient of H is the trivial one. We conclude the proof by observing that every finitely generated group has a maximal normal subgroup by Zorn’s lemma, hence a finitely generated simple quotient.  Remark 2.3 As observed by G. Baumslag [9], Lemma 2.1 also implies that the one-relator group B = a, b | bab−1 aba−1 b−1 = a2 , which is infinite and non-cyclic by Magnus’ Freiheitssatz, has all its finite quotients cyclic. The Higman group H itself is far from simple: indeed, it is SQ-universal by [112, Corollary 3]. The interest for the Higman group remains vivid in contemporary research: we refer to A. Martin’s recent work [86] for a description of its intriguing geometric features. 2.2

The first explicit family, after R. Camm

Shortly after Higman’s theorem 2.2 was published, R. Camm obtained the first explicit example of a finitely generated infinite simple group. Her construction actually yields uncountably many pairwise non-isomorphic such groups. Theorem 2.4 (R. Camm [24]) There is an explicit uncountable set R of triples of permutations of the non-zero integers such that for all (ρ, σ, τ ) ∈ R, the group C = a, p, b, q | ai pρ(i) = bτ (i) q στ (i) , i ∈ Z \ {0} is an infinite torsion-free simple group. The fact that C is infinite and torsion-free is clear since C is a free amalgamated product of two free groups of rank 2 over a common infinitely generated subgroup. In particular C is not finitely presentable (see [89, Corollary 12]). 2.3

The first finitely presented infinite simple group, after R. Thompson

In an unpublished manuscript that was circulated in 1965, R. J. Thompson introduced three finitely generated infinite groups, known as Thompson’s groups F , T and V . Thompson showed that T and V are finitely presented and simple; they were the first known specimen of that kind, see [25]. G. Higman [61] showed that Thompson’s group V is a member of an infinite family of finitely presented infinite simple groups, known as the Higman–Thompson groups. The following result records K. Brown’s presentation of Thompson’s group V as the fundamental group of a triangle of finite groups (we refer to [17, §II.12] for the basic theory of simple complexes of groups). Other short presentations of V were also found more recently by Bleak–Quick [14].

Caprace: Finite and infinite quotients of discrete and indiscrete groups

21

Theorem 2.5 (K. Brown [18, Theorem 3]) The infinite simple Thompson group V is isomorphic to the fundamental group of a triangle of finite groups, whose vertex groups are isomorphic to Sym(5), Sym(6) and Sym(7) respectively. Triangles of finite groups constitute the second simplest kind of amalgams of finite groups after finite graphs of finite groups. Fundamental groups of the latter are all virtually free groups, hence residually finite. That the fundamental group of a triangle of finite groups may fail to be residually finite is rather striking. As observed by K. Brown [18], his Theorem 2.5 provided a negative answer to a question of B. H. and H. Neumann [90]. It is also natural to ask whether the fundamental group of a triangle of finite groups must be residually finite, or whether it can be simple, under the additional hypothesis that it has non-positive curvature in the sense of [17, §II.12]. We emphasize that Brown’s triangle of groups from Theorem 2.5 fails to satisfy that curvature hypothesis (see the Remark following Theorem 3 in [18]); in fact Thompson’s group V contains a copy of every finite group, and therefore cannot have a proper cocompact action on a CAT(0) space. The existence of a non-positively curved triangle of finite groups with a non-residually finite fundamental group has been proved by Hsu–Wise [63]. However, the following question remains open.

Problem 2.6 Can the fundamental group of a non-positively curved triangle of finite groups be simple? Explicit candidates for a positive answer may be found among the 48 triangles of finite groups discussed by J. Tits in his contribution to the proceedings of the Groups St Andrews 1985 conference, [119, §3.1]. Those can be constructed as follows. Up to isomorphism, there are exactly four triangles of groups whose vertex groups are Frobenius groups of order 21, and whose edge groups are cyclic of order 3; this was first observed by M. Ronan [109, Theorem 2.5]. Similarly, there are exactly 44 isomorphism classes of triangles of finite groups with edge groups C9 cyclic of order 9 and vertex groups isomorphic to the Frobenius group C73  C9 , see [119, §3.1] and [120, §3.2 in Cours 1984–85]. The fundamental group of each of them acts simply transitively on the chambers of a Euclidean building of type A˜2 . Therefore, all 48 of them are hereditarily just-infinite by an unpublished theorem of Y. Shalom and T. Steger. A couple of them are linear (in characteristic 0 and 2 respectively) by [72, 74], while most do not admit any finite-dimensional representation over any commutative unital ring by [6, Theorem 1.1 and §1.3]. It is conjectured that all of those non-linear groups fail to be residually finite, and are thus virtually simple by Proposition 1.1. However, none of the 4 smaller triangles have a simple fundamental group by [73], but it is possible that the fundamental group of one of the 44 larger ones is so. Indeed, among those 44 groups, 22 are perfect. Computer experiments led by Stefan Witzel with MAGMA showed that one of these 22 groups has PSL2 (F73 ) as a quotient, while none of the other 21 groups admits any finite simple quotient of order < 2.1010 .

22 2.4

Caprace: Finite and infinite quotients of discrete and indiscrete groups Quotients of free amalgamated products of free groups

The knowledge of finitely generated simple amalgams of free groups on one hand (see Theorem 2.4), and of finitely presented infinite simple groups on the other hand (see Theorem 2.5) led P. Neumann to ask the following question in 1973. Question 2.7 (P. Neumann [91, Problem in §3]) Can an amalgamated free product G = A ∗C B with A, B finitely generated free and C of finite index in A and B, be a simple group? Or is it always SQ-universal? Notice that an amalgamated free product of that kind is finitely presented and torsion-free. The necessity of imposing that the index of C be finite in A and B comes from the following (see also [87] for more general results on the SQuniversality of groups acting on trees). Theorem 2.8 ([112, Corollary 2]) Let G = A ∗C B be an amalgamated free product of groups A, B, C, where A and C are finitely generated free groups. If [A : C] is infinite and B = C, then G is SQ-universal. P. Neumann’s question generated a substantial amount of research, and was eventually answered by M. Burger and S. Mozes [22], see Section 4 below. Before presenting their solution, we mention the following result of M. Bhattacharjee, providing examples of finitely presented amalgamated products of free groups whose only finite quotient is the trivial one. Theorem 2.9 (M. Bhattacharjee [13, Theorem 8]) There exists an amalgamated free product G = A ∗C B, where A and B are free groups of rank 3 and C is a free group of rank 13 embedded as a subgroup of index 6 in A and B, whose only finite quotient is the trivial one. M. Battacharjee provides a very explicit presentation of that amalgam. At the time of this writing, it is still unknown whether Bhattacharjee’s group is simple or not, or even if the amalgam is faithful1 (i.e., the G-action on the Bass–Serre tree of the amalgam is faithful). In order to control infinite quotients of certain free amalgamated products of free groups, M. Burger and S. Mozes elaborated on ideas originally developed by G. Margulis in his seminal work on discrete subgroups of semi-simple Lie and algebraic groups. We shall now give a brief introduction to the relevant material, based on the discussion of specific families of examples. 1 We anticipate the terminology that will be introduced in Section 4 to make the following comment. Among the defining relations of the group considered by Battacharjee, there is a relation specifying that one of the generators is conjugate to its square. Since the group defined by Battacharjee’s presentation is torsion-free, that relation prevents the group from acting properly cocompactly on a CAT(0) space, since an element conjugate to its square must have zero translation length, hence be torsion by [17, Proposition II.6.10(2)]. In particular the Battacharjee group is not commensurable to any BMW-group.

Caprace: Finite and infinite quotients of discrete and indiscrete groups

3 3.1

23

Kneser’s simplicity conjecture The multiplicative group of the Hamiltonian quaternions

Given a commutative domain R of characteristic = 2, we denote by H(R) = {x0 + x1 i + x2 j + x3 k | xi ∈ R} the ring of Hamiltonian quaternions over R. The symbols i, j, k are subjected to the usual multiplication rules: i2 = j 2 = k 2 = −1 and ij = k = −ji. The conjugate of a quaternion x = x0 + x1 i + x2 j + x3 k is defined as x ¯ = x0 − x1 i − x2 j − x3 k and the reduced norm of x is the product Nrd(x) = x¯ x, which is an element of R. Over the field of real numbers R, we get the standard Hamiltonian quaternions, which form a division algebra over R. Similarly H(K) is a division algebra for any subfield K ⊂ R. We denote its multiplicative group by H(K)∗ . It is easy to check that the center of the group H(K)∗ consists of the non-zero scalars x0 ∈ K ∗ . It can also be seen that the quotient group H(K)∗ /K ∗ is infinitely generated for any subfield K ⊂ R; this will become apparent later on. Observe that the reduced norm Nrd : H(K)∗ → K ∗ is multiplicative. Its kernel is denoted by SL1 (H(K)). The center of SL1 (H(K)) is the cyclic group {±1} of order 2, and the embedding of SL1 (H(K)) in H(K)∗ induces an injective homomorphism SL1 (H(K))/{±1} → H(K)∗ /K ∗ . To describe its image, observe that the reduced norm descends to a homomorphism ν : H(K)∗ /K ∗ → K ∗ /(K ∗ )2 whose target is an abelian group of exponent 2. The kernel of ν coincides with the image of the embedding SL1 (H(K))/{±1} → H(K)∗ /K ∗ . It consists of those cosets xK ∗ represented by a non-zero quaternion x whose reduced norm Nrd(x) is a square in K ∗ . We recall that over the complex numbers, the algebra H(C) is isomorphic to the matrix algebra M2 (C); an isomorphism is given by the map   x0 + x1 i x 2 + x3 i x0 + x1 i + x2 j + x3 k → . −x2 + x3 i x0 − x1 i In particular, for any subfield K ⊂ R, the group H(K)∗ /K ∗ embeds as a subgroup of PGL2 (C). This map identifies H(R)∗ /R∗ with the compact Lie group PU(2). Since the reduced norm of a quaternion x ∈ H(R)∗ is a positive real number, it is a square in R∗ , hence the embedding SL1 (H(R))/{±1} → H(R)∗ /R∗ is surjective. This yields the isomorphism PSU(2) ∼ = PU(2). Recall moreover that the latter group is simple. What is the normal subgroup structure of the group SL1 (H(K))/{±1} for other subfields K ⊂ R? Our goal in this section is to provide a partial answer to that question by discussing the following facts, which are special cases of a result due to G. Margulis.

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Caprace: Finite and infinite quotients of discrete and indiscrete groups

Theorem 3.1 (G. Margulis [84]) Let K ⊂ R be a number field. (i) The group SL1 (H(K))/{±1} is hereditarily just-infinite. (ii) If for every non-archimedean local field k containing K as a subfield, the algebra H(k) is isomorphic to M2 (k), then the group SL1 (H(K))/{±1} is simple; in particular the derived group of H(K)∗ /K ∗ is simple. Otherwise it is residually finite. Theorem 3.1(ii) is a special case of a theorem from [84], whose scope encompasses the reduced norm 1 group SL1 (D)/{±1} for all quaternion division algebras D over an arbitrary number field. That simplicity statement was known as Kneser’s conjecture, in reference to Kneser’s remark following Satz C in [71]. We refer the reader to [98, Chapter 9] for a detailed account on this fascinating subject; see also Section 3.3 below for a brief discussion of the tremendous amount of research that it triggered. The point we would like to make here is that Margulis’ proof of the simplicity part in Theorem 3.1(ii) relies on (i) in an essential way. In other words, the proof of simplicity is achieved in two steps: the first is to exclude non-trivial infinite quotients, and was achieved by Margulis using his celebrated Normal Subgroup Theorem [83], to which the next section is devoted. The second step is to analyze the finite quotients. As indicated by the statement, the nature of those finite quotients happens to depend on the arithmetic properties of K. Let us illustrate this matter of facts by concrete examples of number fields. To that end, let us recall that for any local field k of residue characteristic p = 2, the Hamiltonian quaternion algebra H(k) is isomorphic to the matrix algebra M2 (k), see [65, Proposition 9.14]. This is however not true if k has residue characteristic 2: indeed, it can be checked that H(Q2 ) is a division algebra, see [65, Exercise 5 in Chapter 9]. Now, if H(k) is a division algebra, then the reduced norm 1 group SL1 (H(k)) is compact when endowed with the topology induced by the ultrametric topology on k (this can checked explicitly; for a more general fact see [98, Theorem 3.1]). Moreover it is totally disconnected, because the topology of k is so. In particular it is a profinite group. If the number field K embeds in such a local field k, then SL1 (H(K))/{±1} embeds in the profinite group SL1 (H(k))/{±1}, and is thus residually finite. This is in particular the case for K = Q since H(Q2 ) is a division algebra. √ On the other hand 2 is not an element of Q2 , since otherwise its 2-adic valuation would be 1/2, which is impossible because the 2-adic valuation of any element of √ Q2 is an integer. Thus Q2 ( 2) is a quadratic extension of Q2 . Therefore it follows √ from [65, Proposition 9.13] that H(Q ( 2)) is isomorphic to the matrix algebra 2 √ M2 (Q2 ( 2)). Any local field k of residue characteristic 2 that contains a copy √ √ √ of K = Q( 2) also contains Q ( 2) (namely Q ( 2) is the closure of K in k). 2 2 √ We deduce√that K = Q( 2) satisfies the condition of Theorem 3.1(ii), so that SL1 (H(Q( 2)))/{±1} is simple. For details on the proof of Theorem 3.1(ii), we refer to [98, Chapter 9]. We shall now emphasize the relevance of non-discrete locally compact groups in the proof

Caprace: Finite and infinite quotients of discrete and indiscrete groups

25

of Theorem 3.1(i). 3.2

The Margulis Normal Subgroup Theorem

A lattice in a locally compact group G is a discrete subgroup Γ ≤ G such that the quotient space G/Γ carries a G-invariant probability measure. For an excellent treatment of the basic theory of lattices, we refer to [102]. A detailed exposition of the following fundamental result, first established in [83], may be consulted in [85, §IV.4 and §IX.5]. Given a product group G1 × · · · × Gn and a subset A ⊆ {1, . . . , n} of the index set, we denote by GA the subproduct GA = i∈A Gi , that we identify in the natural way with a direct factor of G1 × · · · × Gn . Theorem 3.2 (Margulis Normal Subgroup Theorem [85, Theorem (4) in the Introduction]) Let n ≥ 1 and for each i = 1, . . . , n, let ki be a non-discrete locally compact field, let Gi be an almost ki -simple algebraic group over ki , let ri be the ki -rank of Gi , and assume that ri > 0. Let Gi be the quotient of Gi (ki ) by its center. Let Γ < G1 × · · · × Gn be a lattice. Assume that for every partition {1, . . . , n} = A ∪ B with A = ∅ = B, the product group (Γ ∩ GA )(Γ ∩ GB ) is of infinite index in Γ. If ni=1 ri ≥ 2, then Γ is hereditarily just-infinite. The hypothesis that (Γ ∩ GA )(Γ ∩ GB ) is of infinite index in Γ expresses the fact that Γ is an irreducible lattice. That condition is obviously necessary for Γ to be hereditarily just-infinite, since the intersections Γ∩GA and Γ∩GB are both normal subgroups of Γ. The hypothesis on the rank of G1 × · · · × Gn is also necessary: indeed lattices in rank 1 simple Lie groups are never just-infinite by [54]. Actually they are all SQ-universal, see Theorem 5.6 below. Let us illustrate Theorem 3.2 with a specific example related to the previous section. We retain the notation introduced there. 1 Given a finite set of primes S = { 1 , . . . , r }, we consider the ring ZS = Z[ 1 ... ] r consisting of those rationals x ∈ Q whose p-adic valuation is non-negative for all primes p ∈ S. We also let LS = H(ZS )∗ x ¯ be the multiplicative group of units of the ring H(ZS ). Since x−1 = Nrd(x) for all ∗ x ∈ H(Q) , we see that an element x ∈ H(ZS ) is a unit if and only if its reduced norm Nrd(x) is a unit of the ring ZS . As in the previous section, the prime 2 and the odd primes play a different role. We first consider p = 2. Clearly, any x ∈ L{2} can be written as 2n y, where n ∈ Z and y ∈ H(Z). By [41, Lemma 2.5.5], any quaternion y ∈ H(Z) can be written as a product y = 2m πε for some m ∈ N, some π ∈ {1, 1 + i, 1 + j, 1 + k, (1 + i)(1 + j), (1 + i)(1 − k)} and some ε ∈ H(Z) whose reduced norm is odd. Since Nrd(x) ∈ Z[ 12 ]∗ , we deduce that ε belongs to L∅ = H(Z)∗ = {±1, ±i, ±i, ±j, ±k}. It follows that the group L{2} /Z[ 12 ]∗ is finite. On the other hand, for any odd prime p, the group L∗{p} /Z[ p1 ]∗ is infinite and finitely generated, see [41, Theorem 2.5.13]. As mentioned in the previous section, if the prime p is odd, the Hamiltonian quaternion algebra H(Qp ) over the p-adic numbers is isomorphic to the matrix

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algebra M2 (Qp ), which yields a natural injective homomorphism ϕp : H(Q)∗ /Q∗ → PGL2 (Qp ). Given a finite set of primes S containing at least one odd prime, let ΓS be the image of the product homomorphism   ϕp : LS /Z∗S → PGL2 (Qp ) = GS . p∈S\{2}

p∈S\{2}

By [122, Chapter IV, Theorem 1.1], the group ΓS is a discrete subgroup of GS and the discreteness is due to the fact that  quotient space GS /ΓS is compact. 1The ΓS ∩ p∈S\{2} PGL2 (Zp ) ≤ Γ{2} ∼ = L{2} /Z[ 2 ]∗ , which is a finite group as mentioned above. Moreover, for each p ∈ S \ {2}, the homomorphism ϕp maps injectively LS /Z∗S to a Zariski dense subgroup of PGL2 (Qp ), so that no finite index subgroup of LS /Z∗S splits as the direct product of two infinite subgroups. Those facts imply that for any non-empty finite set of primes S containing at least two odd primes, the hypotheses of Theorem 3.2 are satisfied, so that the group ΓS is hereditarily just-infinite. Since the group H(Q)∗ /Q∗ is the directed union of the collection of subgroups LS /Z∗S indexed by the finite sets of primes S, it readily follows from Theorem 3.2 that every proper quotient of H(Q)∗ /Q∗ is a locally finite group, i.e., a group in which every finitely generated subgroup is finite. Similarly, every proper quotient of the subgroup SL1 (H(Q))/{±1} is locally finite. In fact, as observed by Margulis [83], using Strong Approximation one can show via Theorem 3.2 that SL1 (H(Q))/{±1} is hereditarily just-infinite (see [98, p. 517]). This is how the proof of Theorem 3.1(i) is completed in the case of K = Q. Remark 3.3 If S = {p, } is a set consisting of two distinct odd primes, then the group ΓS is a cocompact lattice with dense projections in PGL2 (Qp ) × PGL2 (Q ). Using the actions of PGL2 (Qp ) and PGL2 (Q ) on their associated Bruhat–Tits trees, one can show that ΓS has a finite index subgroup which splits as an amalgamated free product A ∗C B where A and B are finitely generated free groups and C is of finite index in A and B. We refer to Proposition 4.5 below for a concrete example with S = {3, 5}. Since ΓS is hereditarily just-infinite, this shows that an amalgam as in Question 2.7 can be hereditarily just-infinite; in particular it may fail to be SQ-universal. 3.3

Finite quotients of the multiplicative group of a division algebra

Kneser’s conjecture has been successively generalized by V. Platonov [97] and G. Margulis [83]; its most general form, formulated in [83, §2.4.8], is known as the Margulis–Platonov conjecture. That conjecture describes the normal subgroup structure of the group of rational points of a simply connected simple algebraic group over a number field. Although the conjecture is still open in full generality, the special case of the reduced norm 1 group SL1 (D) of a division algebra D over a number field was settled in a series of important papers by various authors (see [114], [103] and references therein). Roughly speaking, the normal subgroup structure of SL1 (D) is elucidated by a similar scheme as in the proof of Theorem 3.1:

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27

infinite quotients and finite quotients are investigated separately, with completely different methods. While the treatment of infinite quotients is based on Margulis’ Normal Subgroup Theorem and Strong Approximation as above, the Classification of the Finite Simple Groups (CFSG) is used in an essential way to investigate the finite quotients of SL1 (D) for all division algebras D of degree ≥ 3 over number fields, see [104], [114] and [103]. We finish this section by mentioning the following striking culmination of this direction of research, which is valid over an arbitrary ground field. The proof relies on the CFSG via the consideration of commuting graphs. Theorem 3.4 (Rapinchuk–Segev–Seitz [105]) Let D be a division algebra which is finite-dimensional over its center. Every finite quotient of the multiplicative group D∗ is solvable.

4

Lattices in products of trees, after M. Burger, S. Mozes and D. Wise Let’s start with the A-B-C of it Getting right down to the X-Y-Z of it Help me solve the mystery of it G. De Paul, S. Cahn, Teach me tonight, 1953

The goal of this section is to discuss a class of discrete groups acting properly and cocompactly on a product of regular locally finite trees. The study of those groups was pioneered by S. Mozes [88], Burger–Mozes [20, 21, 22] and D. Wise [130] in the mid 1990’s, and provided notably an answer to P. Neumann’s Question 2.7. 4.1

BMW-groups

We recall that the Cartesian product of two graphs (V1 , E1 ) and (V2 , E2 ) is the graphwhose vertex set is  V1 × V2 and whose edge set is defined as the collection of pairs {v1 , v2 }, {w1 , w2 } such that either {v1 , w1 } ∈ E1 and v2 = w2 , or {v2 , w2 } ∈ E2 and v1 = w1 . A group Γ is called a BMW-group if Γ is capable of acting by automorphisms on the Cartesian product of two trees, say T and T  , so that every element of Γ preserves the product decomposition T × T  (i.e., no element of Γ swaps the factors T and T  ), and that the action of Γ on the vertex set of T ×T  is free and transitive. Equivalently, Γ is a BMW-group if it has a finite generating set Σ such that the Cayley graph of (Γ, Σ), viewed as an undirected unlabeled graph, is isomorphic to the Cartesian product of two trees, say T and T  , and if the image of the associated homormorphism Γ → Aut(T × T  ) is contained in Aut(T ) × Aut(T  ). Since Γ acts vertex-transitively on its Cayley graphs, it follows that T and T  must be regular trees. The degree of a BMW-group is the pair (deg T, deg T  ), which depends a priori on the choice of the generating set Σ. Recall that the groups Γ admitting a finite generating set Σ such that the Cayley graph of (Γ, Σ) is a tree, are the free products of finitely generated free groups and

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finitely generated free Coxeter groups. A free Coxeter group is a free product of cyclic groups of order 2. We emphasize that two non-isomorphic groups can have the same tree as a Cayley graph. This matter of fact is greatly amplified when passing from single trees to Cartesian products of two trees. Indeed, as we shall see, the various BMW-groups admitting a given product of trees as a Cayley graph can enjoy radically different algebraic properties. The product C2 × C2 of two cyclic groups of order 2 is the only BMW-group of degree (1, 1). For all m, n, the direct product of the free Coxeter group of rank m and the free Coxeter group of rank n is a BMW-group of degree (m, n). Similarly, the direct product of the free group of rank m and the free group of rank n is a BMW-group of degree (2m, 2n). A BMW-group is called reducible if it has a finite index subgroup that splits as a direct product of two free groups of rank ≥ 1. The following example of a BMW-group was studied by D. Wise in his thesis [130], where it is notably proved to be irreducible. Example 4.1 The group ΓWise = a, b, x, y, z | aya−1 x−1 , byb−1 x−1 , azb−1 z −1 , axb−1 y −1 , bxa−1 z −1 , bza−1 y −1 is an irreducible BMW-group of degree (4, 6). It is called the Wise lattice. A key property of BMW-groups is that they can be identified by means of a presentation of a very specific form, defined as follows. A BMW-presentation is a group presentation of the form Γ = A ∪ X | R , where A and X are disjoint finite sets, and the set of relations R satisfies the following two conditions: (BMW1) R has a (possibly trivial) partition R = R2 ∪ R4 , such that every r ∈ R2 is of the form r = t2 with t ∈ A ∪ X, and every r ∈ R4 is of the form r = axa x with a, a ∈ A ∪ A−1 and x, x ∈ X ∪ X −1 ; (BMW2) For all a ∈ A ∪ A−1 and x ∈ X ∪ X −1 , there exists a unique a ∈ A ∪ A−1 and a unique x ∈ X ∪ X −1 such that axa x or a x ax or a−1 x a x−1 or a x−1 a−1 x belongs to R4 . To interpret correctly the uniqueness conditions appearing in (BMW2), it is important to view the elements of R4 as words in the group A ∪ X | R2 . Hence, if (a )2 ∈ R2 , then a and (a )−1 are the same element of A ∪ A−1 , and similarly for x . The next result collects basic properties of BMW-groups and BMW-presentations. The proof uses basic results on CAT(0) groups as well as standard tools from Bass–Serre theory (some details may be found in [22, Section 6.1], [106, Chapter I] and [99, §3–4]).

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Proposition 4.2 Every BMW-group admits a BMW-presentation. Conversely, let Γ = A ∪ X | R be a BMW-presention, with R = R2 ∪ R4 as above. Let A = {a ∈ A | a2 ∈ R2 }, X  = {x ∈ X | x2 ∈ R2 } and let m = |A \ A |, m = |A |, n = |X \ X  | and n = |X  |. (i) The Cayley graph of Γ with respect A ∪ X is isomorphic to the Cartesian product TA × TX of two regular trees of degree M = 2m + m and N = 2n + n respectively. In particular Γ is a BMW-group of degree (M, N ) = (2m + m , 2n + n ). (ii) We have |R4 | ≥ mn. Moreover, if R2 = ∅ and |R4 | = mn then Γ is torsionfree. (iii) Every torsion-free BMW-group of degree (2m, 2n) admits a BMW-presention with |R4 | = mn. (iv) The subgroup A is the free product of a free group of rank m with a free Coxeter group of rank m ; it fixes a vertex in TX and acts simply transitively on the vertices of TA . Similarly X is the free product of a free group of rank n with a free Coxeter group of rank n ; it fixes a vertex in TA and acts simply transitively on the vertices of TX . (v) Γ has a torsion-free normal subgroup Γ+ of index 4 with Γ/Γ+ ∼ = C2 × C2 . The group Γ+ acts without edge-inversion on both TA and TX . (vi) If the Γ+ -action on TA is edge-transitive, then Γ+ admits a decomposition as a free amalgamated product of the form Γ+ ∼ = FN −1 ∗FM N −2M +1 FN −1 , where Fd denotes the free group of rank d. Similarly, if the Γ+ -action on TX is edge-transitive, then Γ+ admits a decomposition as a free amalgamated product of the form Γ+ ∼ = FM −1 ∗FM N −2N +1 FM −1 . (vii) Given a BMW-presentation Γ = A ∪ X  | R with A ⊂ A , X ⊂ X  and R ⊂ R , the natural homomorphism Γ → Γ induced by the inclusion of the generating set of Γ is injective. Moreover the Cayley graph Cay(Γ, A ∪ X) embeds as a Γ-invariant convex subgraph of Cay(Γ , A ∪ X  ). The presentation 2-complex of a torsion-free BMW-group Γ of degree (2m, 2n) with BMW-presentation Γ = A ∪ X | R is called a BMW-complex of degree (2m, 2n). It is a square complex Y with a single vertex v, m + n oriented edges labeled by the m elements of A and the n elements of X, and mn squares corresponding to the relations in R. The condition (BMW2) corresponds to the geometric property that the link of Y at v is isomorphic to the complete bipartite graph K2m,2n . The universal cover of Y is isomorphic to the product T2m × T2n of the regular trees of degrees 2m and 2n, viewed as a square complex. The group Γ is isomorphic to the fundamental group π1 (Y ); it acts on T2m × T2n by covering transformations, and the action preserves each of the two tree factors.

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4.2

Examples of BMW-groups of small degree

We now describe some BMW-groups of degree (M, N ) for the smallest values of M and N . We start with the torsion-free case. Up to isomorphism, the only torsion-free BMW-groups of degree (2, 2) are the free abelian group Z2 = a, x | axa−1 x−1 and the Klein Bottle group a, x | axa−1 x . For every n, all torsion-free BMW-groups of degree (2, 2n) are reducible. This follows from Theorem 4.9 below. The torsion-free BMW-groups of degree (4, 4) have been studied by Kimberley– Robertson [69] and D. Rattaggi [106]. As explained in Section 7 from [69], there are exactly 52 homeomorphism types of BMW-complexes of degree (4, 4). It is important to underline that two non-homeomorphic complexes can have isomorphic fundamental groups. The number of isomorphism classes of torsion-free BMWgroups of degree (4, 4) is not known, but according to loc. cit. it belongs to the set {41, 42, 43}. By comparing and combining Table C.4 on p. 278 in Section C.5 of [106] with Section 7 from [69] (keeping an eye on the structure of the abelianization), one can see that among the 52 BMW-complexes of degree (4, 4), at least 50 have a reducible fundamental group. As we shall see, the remaining two complexes happen to have an irreducible fundamental group. Those admit the following BMW-presentations. Example 4.3 The groups ΓSV = a, b, x, y | axay, ax−1 bx−1 , ay −1 b−1 y −1 , bxby −1 and

ΓJW = a, b, x, y | axay, ax−1 by −1 , ay −1 b−1 x−1 , bxb−1 y −1

are the only two irreducible torsion-free BMW-groups of degree (4, 4), up to isomorphism. The irreducibility of ΓSV is a consequence of the main results in [117]. Proposition 4.4 (J. Stix and A. Vdovina [117]) The BMW-group ΓSV is irreducible. It embeds as a cocompact lattice with dense projections in PGL2 (F3 ((t)))× PGL2 (F3 ((t))). In particular it is hereditarily just-infinite by the Margulis Normal Subgroup Theorem. The fact that ΓJW is irreducible is established by Janzen–Wise in [66]. Although the presentations of ΓSV and ΓJW are rather similar, the groups are quite different. Indeed ΓSV is linear in characteristic 3, while the group ΓJW fails to be residually finite (this was observed independently in [16, Theorem 15] and [34]; see also Section 4.6 below). One can pursue the enumeration of BMW-complexes of larger degrees. The Wise lattice is an example of an irreducible torsion-free BMW-group of degree (4, 6). As we shall see in Section 4.6 below, it is not residually finite. Another example of degree (4, 6) is provided by the following result, due to D. Rattaggi.

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Proposition 4.5 (D. Rattaggi [106, Theorem 3.35 and Proposition 3.47]) The BMW-group ΓRatt = a, b, x, y, z | axby, aybx−1 , azb−1 x, az −1 ay −1 , ax−1 b−1 z, bzby −1 is irreducible. It embeds as a cocompact lattice with dense projections in PGL2 (Q3 ) × PGL2 (Q5 ). In particular it is hereditarily just-infinite by the Margulis Normal Subgroup Theorem. The lattice ΓRatt happens to be a quaternionic arithmetic lattice as those discussed in Section 3.2. More precisely, consider the quaternion algebra D = H(Q) with standard basis {1, i, j, k}. One can compute that the assignments: ϕ(a) ϕ(b) ϕ(x) ϕ(y) ϕ(z)

= = = = =

1 − i − j, 1 − i + j, 1 + 2k, 1 − 2i, 1 − 2j

extend to a homomorphism ϕ : ΓRatt → D∗ /Q∗ , by checking that the defining relations of ΓRatt are satisfied. It turns out that ϕ is injective, and maps ΓRatt to a finite index subgroup of the lattice H(Z{3,5} )∗ /Z∗{3,5} discussed in Section 3.2. The following example of degree (6, 6) was pointed out to me by I. Bondarenko. Proposition 4.6 (I. Bondarenko, D. D’Angeli, E. Rodaro) The BMW-group ΓBDR = a, b, c, x, y, z | axa−1 x−1 , ayb−1 z −1 , azc−1 y −1 , cyc−1 x−1 , bya−1 y −1 , bxc−1 z −1 , bzb−1 x−1 , cxb−1 y −1 , cza−1 z −1 is an irreducible torsion-free BMW-group of degree (6, 6). The image of the respective projections of the free groups a, b, c and x, y, z to the automorphism groups of the tree factors T{x,y,z} and T{a,b,c} are both isomorphic to the lamplighter group C3  Z. Proof The fact that ΓBDR is a torsion-free BMW-group of degree (6, 6) is clear from Proposition 4.2. The statement on the projection of ΓBDR to the automorphism groups of the tree factors follows from [51, Corollary 2.14] and the main result in [15]. The irreducibility follows from the statement on the projections together with Theorem 4.9 below.  We now present some examples of BMW-presentations involving generators of order 2. As remarked in Proposition 4.2, if all generators of a BMW-group Γ in a BMW-presentation are of infinite order, then the degree of Γ consists of a pair of even integers. Allowing the generators to be torsion gives rise to BMW-groups in odd degree.

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Proposition 4.7 (N. Rungtanapirom [110, Theorem A]) The BMW-group ΓRung = a, b, x, y | a2 , x2 , axax, ayby, bxby −1 is irreducible of degree (3, 3). It embeds as a cocompact lattice with dense projections in PGL2 (F2 ((t))) × PGL2 (F2 ((t))). In particular it is hereditarily just-infinite by the Margulis Normal Subgroup Theorem. The following examples, respectively of degree (3, 3), (4, 5) and (6, 6), are due to N. Radu [99]. They all contain non-trivial torsion elements. Proposition 4.8 (N. Radu) The BMW-groups Γ3,3 = a, b, c, x, y, z | a2 , b2 , c2 , x2 , y 2 , z 2 , axax, ayay, azbz, bxbx, bycy, cxcz , Γ4,5 = a, b, c, d, x, y, z, t, u | a2 , b2 , c2 , d2 , x2 , y 2 , z 2 , t2 , u2 , axax, ayay, azbz, bxbx, bycy, cxcz, xtxt, atat, audu, btcu, dxdy, dzdt and Γ6,6 = a, b, c, x, y, z | axay, ax−1 by −1 , ay −1 b−1 x−1 , bxb−1 y −1 , cx−1 c−1 x−1 , c−1 yc−1 y, cycz −1 , az −1 a−1 z, bzcz, bz −1 bz −1 are irreducible of degree (3, 3), (4, 5) and (6, 6) respectively. The subgroup a, b, c, x, y, z of Γ4,5 is isomorphic to Γ3,3 ; the subgroup a, b, x, y of Γ6,6 is isomorphic to the BMW-group ΓJW from Example 4.3. Proof The irreducibility of Γ3,3 is established in [99, Proposition 5.4]; that assertion can actually be deduced from Corollary 4.13 below. That Γ3,3 (resp. ΓJW ) embeds naturally as a subgroup of Γ4,5 (resp. Γ6,6 ) follows from Proposition 4.2(vii). The irreducibility of Γ4,5 and Γ6,6 can then be deduced from the irreducibility of Γ3,3 and ΓJW using Theorem 4.9 below.  The group Γ4,5 is denoted by Γ4,5;9 in [99, §5.2], while Γ6,6 appears as Γ6,6;2 in [99, §5.1]. We will come back to those lattices in Section 4.9 below. Although the generators of Γ6,6 are of infinite order, the elements bz −1 and c−1 y ∈ Γ6,6 have order 2, as follows clearly from the defining relations. It is important to remark that the number of homeomorphism types of BMWcomplexes grows quickly with the degree. A lower bound on that number is given by an explicit formula in [117, Formula (2.4) in §2.3]2 . With the help of a computer, N. Radu [101] has enumerated all BMW-presentations of small degree. In 2 The number appearing in the formula (2.4) in §2.3 of [117] indeed yields a lower bound on the number of those homeomorphism types.

Caprace: Finite and infinite quotients of discrete and indiscrete groups

33

particular, he has shown that there are 1001 BMW-complexes of degree (4, 6). Among them, at least 890 have a reducible fundamental group, while at least 16 are irreducible. Moreover, there are 32062 BMW-complexes of degree (6, 6), among which at least 18426 are reducible and at least 8227 are irreducible, see [101]. In either case, the exact number of irreducible ones is unknown; neither is the number of those with a linear or residually finite fundamental group. The difficulty is that there is no known necessary and sufficient condition determining whether a BMW-group is irreducible (or linear, or residually finite, or just-infinite) that can be checked algorithmically on the BMW-presentation. Problems of that nature are recorded in [132, Section 10]. In the next sections, we shall discuss sufficient (but not necessary!) conditions that can be used to check some of those properties algorithmically. 4.3

Inseparability and irreducibility

A fundamental early discovery of D. Wise [130] is that the irreducibility of a BMWgroup Γ = A ∪ X | R is related to the inseparability of the subgroups A and X in Γ. This phenomenon was first highlighted by him in the case of the Wise lattice in [130] (see also [132, Corollary 6.4]). We shall present a general statement recently established in [29, Corollary 32] and inspired by Wise’s work [132, 131]. The statement requires the following terminology. A subgroup H of a group G is called separable if it is an intersection of finite index subgroups of G. Equivalently H is separable if and only if for every g ∈ G, if g ∈ H then there exists a finite quotient ϕ : G → Q with ϕ(g) ∈ ϕ(H). The set of separable subgroups of G is closed under intersections. The profinite closure of a subgroup H in G, denoted by H, is the smallest separable subgroup of G containing H. It coincides with the closure of H with respect to the profinite topology on G. A subgroup H of G is called virtually normal if it has a finite index subgroup which is normal in G. It is weakly separable if it is an intersection of virtually normal subgroups of G. Equivalently H is weakly separable if and only if for every g ∈ G, if g ∈ H then there exists a (possibly infinite) quotient ϕ : G → Q such that ϕ(H) is finite and ϕ(g) ∈ ϕ(H). Clearly, every separable subgroup is weakly separable, but not conversely: Indeed, in an infinite simple group, any finite subgroup (including the trivial one) is weakly separable but not separable. Theorem 4.9 ([29, Corollary 32]) Let T1 , T2 be locally finite trees without vertices of degree 1, and let Γ ≤ Aut(T1 ) × Aut(T2 ) be a discrete subgroup acting cocompactly on T1 × T2 . Then the following assertions are equivalent. (i) There exists i ∈ {1, 2} such that the projection pri (Γ) ≤ Aut(Ti ) is discrete. (ii) There exists i ∈ {1, 2} and a vertex or an edge y ∈ V Ti ∪ ETi such that the stabilizer Γy is a weakly separable subgroup of Γ. (iii) For all i ∈ {1, 2} and all y ∈ V Ti ∪ ETi , the stabilizer Γy is a separable subgroup of Γ.

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Caprace: Finite and infinite quotients of discrete and indiscrete groups

(iv) The groups K1 = {g ∈ Aut(T1 ) | (g, 1) ∈ Γ} and K2 = {g ∈ Aut(T2 ) | (1, g) ∈ Γ} act cocompactly on T1 and T2 respectively, and the product K1 ×K2 is of finite index in Γ. If any of those conditions is satisfied, we say that Γ is reducible. Otherwise it is called irreducible. In the special case where Γ is a BMW-group, this terminology coincides with the notion of (ir)reducibility introduced above. The equivalence between (i) and (iv) is due to M. Burger and S. Mozes [22, Proposition 1.2] and follows from general results on lattices in locally compact groups. The fact that the separability of the edge-stabilizers is equivalent to the reducibility of Γ is due to D. Wise, see [131, Lemmas 5.7 and 16.2]. In fact, D. Wise’s results allow one to derive more precise information in the case of BMWgroups. The following assertion will be relevant to our purposes (see [16, Theorem 9] for a related statement). We denote by A∗2 the set of all words of the form ab with a, b ∈ A ∪ A−1 . Thus in a BMW-group Γ = A ∪ X | R , the group A∗2 is the index 2 subgroup of A consisting of the elements of even word length. Proposition 4.10 Let Γ = A ∪ X | R be a BMW-presentation of an irreducible BMW-group. Then the profinite closure A∗2 contains an element of the form xy −1 , for some distinct x, y ∈ X ∪ X −1 . Similarly the profinite closure X ∗2 contains an element of the form ab−1 , with a = b and a, b ∈ A ∪ A−1 . Proof The arguments in the proof of [132, Corollary 6.4] (or alternatively the combination of Lemmas 5.5, 5.7 and 16.2 in [131]) imply that if Γ is irreducible, then for any finite quotient ϕ : Γ → Q, there exist distinct elements x, y ∈ X ∪ X −1 such that ϕ(xy −1 ) ∈ ϕ(A∗2 ). Let us now consider the set P of all pairs of distinct elements x, y ∈ X ∪ X −1 such that there is a finite group Q(x,y) and a homomorphism ϕ(x,y) : Γ → Q(x,y) with ϕ(x,y) (xy −1 ) ∈ ϕ(x,y) (A∗2 ). Denoting by Q the direct product of the groups Q(x,y) , and by ϕ the product homormorphism of the ϕ(x,y) , taken over all (x, y) ∈ P , we obtain a homormorphism ϕ : Γ → Q of Γ to a finite group, such that ϕ(xy −1 ) ∈ ϕ(A∗2 ) for all (x, y) ∈ P . In view of the preceding paragraph, there exists distinct elements x, y ∈ X ∪ X −1 such that (x, y) ∈ P . The conclusion follows. A similar argument may be applied exchanging the roles of A and X.  We emphasize that A ∩X = {1}, see Proposition 4.2, so that Proposition 4.10 indeed witnesses the inseparability of A and X in Γ. 4.4

Anti-tori and irreducibility

Another discovery of D. Wise [130, Section II.4] is that the irreducibility of a BMWgroup Γ = A ∪ X | R can sometimes be established by highlighting what he called

Caprace: Finite and infinite quotients of discrete and indiscrete groups

35

an anti-torus in Γ. By definition, an anti-torus in a group Γ of automorphisms of a product T1 × T2 of two trees is a pair g1 , g2 ∈ Γ preserving the product decomposition and satisying the following conditions: (AT1) For i = 1, 2, there is a point pi ∈ Ti and a geodesic line i ⊂ Ti containing pi such that 1 × {p2 } is invariant under g1 and {p1 } × 2 is invariant under g2 . (AT2) No non-zero powers of g1 and g2 commute. The choice of terminology is motivated by considering the natural CAT(0) realization of the product T1 × T2 . Given a anti-torus {g1 , g2 } in Γ, the lines 1 × {p2 } and {p1 } × 2 span a flat plane whose stabilizer in Γ does not act cocompactly. In other words that flat is not periodic. The following result is due to D. Wise. Proposition 4.11 Let T1 , T2 be locally finite trees without vertices of degree 1, and let Γ ≤ Aut(T1 ) × Aut(T2 ) be a discrete subgroup acting cocompactly on T1 × T2 . If Γ contains an anti-torus, then it is irreducible. Proof See [107, Proposition 9] for a proof in the context of torsion-free BMWgroups, and [66, Lemma 18] for a proof in the more general context of torsion-free lattices in products of trees. Alternatively one may establish the claim without any requirement of torsion-freeness on Γ as follows. Given an anti-torus {g1 , g2 } in Γ, let pi and i be as in (AT1). Then g1 fixes p2 and g2 fixes p1 . The condition (AT2) implies that the projections of g1 on Aut(T2 ) and of g2 on Aut(T1 ) cannot both have finite order. It follows that the projection of Γp1 on Aut(T1 ) or the projection of Γp2 on Aut(T2 ) has infinite image, and is thus non-discrete. Therefore Γ is irreducible by Theorem 4.9.  I do not know whether the converse assertion holds. This amounts to asking the following: Given a cocompact lattice Γ ≤ Aut(T1 ) × Aut(T2 ) and vertices v1 ∈ V (T1 ) and v2 ∈ V (T2 ), is it possible that the projection of the stabilizer Γv1 to Aut(T1 ) and the projection of Γv2 to Aut(T2 ) are both infinite torsion groups? 4.5

Local actions and irreducibility

By Theorem 4.9, in order to prove the irreducibility of a BMW-group, it suffices to show that its projection to the automorphism group of one of the tree factors of its Cayley graph is non-discrete. An idea developed by M. Burger and S. Mozes [21, 22] in order to check that condition is to use some of the geometric aspects of finite group theory. Let us describe that in detail. Let g = (V, E) be a connected locally finite (undirected, unlabeled, simple) graph and Γ ≤ Aut(g) be a group of automorphisms of g. Given a vertex v ∈ V , the local action of Γ at v is the finite permutation group induced by the action of the stabilizer Γv on the sphere S(v, 1) of radius 1 around v. Given an integer r ≥ 0 and a vertex v ∈ V , we set Γ[r] Γw . v = w∈V, d(v,w)≤r

36

Caprace: Finite and infinite quotients of discrete and indiscrete groups [r]

Thus Γv is the pointwise stabilizer of the r-ball around v. The proof of the following fundamental result relies on the classification of the finite 2-transitive groups, which relies in turn on the CFSG. Theorem 4.12 Let Γ ≤ Aut(g) be a vertex-transitive automorphism group of a connected locally finite graph g. Let {v, w} be an edge of g. Suppose that the local action at v is 2-transitive. If the stabilizer Γv is finite, then: (i) (Trofimov–Weiss [121, Theorem 1.4]) We have [5] Γ[5] v ∩ Γv = {1}. [6]

In particular Γv = {1}. [1]

[1]

(ii) (Trofimov–Weiss [121, Theorem 1.3]) If Γv ∩ Γw = {1}, then the local action at v contains a normal subgroup isomorphic to PSLn (Fq ) in its natural action on the points of the n − 1-dimensional projective space over Fq . (iii) (R. Weiss [124, Theorem 1.1]) If the local action at v contains a normal subgroup isomorphic to PSL2 (Fq ) in its natural action on the points of the projective line over Fq , then [3] Γ[3] v ∩ Γw = {1}. [4]

In particular Γv = {1}. The hypothesis that the stabilizer Γv be finite is equivalent to the condition that Γ is a discrete subgroup of Aut(g) endowed with the topology of pointwise convergence on the vertex set V (g), which is second countable, totally disconnected and locally compact. Notice that for any vertex-transitive automorphism group Γ of a connected locally finite graph g, and for any n ≥ 0, we have Γ[n] v = {1}

if and only if

[n+1] Γ[n] . v ≤ Γv

Thus Theorem 4.12 has the following discreteness criterion as an immediate consequence. Corollary 4.13 Let Γ ≤ Aut(g) be a vertex-transitive automorphism group of a connected locally finite graph g. Suppose that the local action at v is 2-transitive. If any of the following conditions holds, then Γ is an indiscrete subgroup of Aut(g): [6]

[7]

[2]

[3]

[4]

[5]

(i) Γv ≤ Γv . (ii) Γv ≤ Γv and the local action of Γ at v is not isomorphic to the action of a linear or semi-linear group of degree n over Fq on the points of the projective n − 1-space over Fq . (iii) Γv ≤ Γv and the local action of Γ at v is isomorphic to the action of a linear or semi-linear group of degree 2 over Fq on the points of the projective line over Fq .

Caprace: Finite and infinite quotients of discrete and indiscrete groups

37

Given a BMW-presentation Γ = A ∪ X | R and associated Cayley graph TA × TX , the group Γ acts vertex-transitively on TA × TX . Moreover, that action preserves the product decomposition. By projecting, we obtain a vertex-transitive action of Γ on TA and another one on TX . In order to apply the criteria from Corollary 4.13, we need to compute the local action of Γ at the base vertex of TA or TX . This can be done effectively as follows. TX axa

a

x 1X

ax

x

1

a

a

x

x 1A

TA

Figure 1. The local action of a on TX at the base vertex 1X The directed edges of the Cayley graph TA × TX are labeled by the elements of A ∪ A−1 ∪ X ∪ X −1 . Consider the base vertex 1 and let 1X be the projection of 1 to the factor TX of the product graph TA × TX , (see Figure 1). Let a ∈ A ∪ A−1 and x ∈ X ∪ X −1 . The element a ∈ A maps the vertex 1 to a; both of those vertices project to 1X in TX . Indeed A fixes the vertex 1X . Moreover a maps the directed edge (1, x) to the directed edge (a, ax), and both edges are labeled by x. Now the presence of the relation axa x in the set R has the following geometric interpretation: the projection of the directed edge (a, ax) to the TX -fiber over 1A in the product decomposition TA × TX coincides with the directed edge (1, (x )−1 ). Thus, viewing TX as a directed graph whose edges are labeled by X ∪ X −1 , we see that the local action of a at 1X maps the outgoing edge at 1X labeled by x to the ingoing edge at 1X labeled by x . Proceeding in that way, one computes that for the BMW-group ΓSV from Example 4.3, the local permutation induced by a and b around the base vertex 1X of the tree TX are a : (x, y −1 , y, x−1 )

and

b : (x, y, y −1 , x)

respectively. Thus the local action of ΓSV at 1X in TX is isomorphic to Sym(4) ∼ = PGL2 (F3 ). By similar arguments, one computes the local actions of the various

38

Caprace: Finite and infinite quotients of discrete and indiscrete groups

examples introduced in the previous section. The information thus obtained is recorded in Table 1.

BMW-group deg(TA )

Local action in TA

deg(TX )

Local action in TX

ΓRung

3

Sym(3) ∼ = PGL2 (F2 )

3

Sym(3) ∼ = PGL2 (F2 )

Γ3,3

3

Sym(3)

3

C2

ΓSV

4

Sym(4) ∼ = PGL2 (F3 )

4

Sym(4) ∼ = PGL2 (F3 )

ΓJW

4

Alt(4) ∼ = PSL2 (F3 )

4

D8

Γ4,5

4

Sym(4)

5

Sym(5)

ΓWise

4

C2 × C2

6

Sym(3) × Sym(3)

ΓRatt

4

Sym(4) ∼ = PGL2 (F3 )

6

PGL2 (F5 )

ΓBDR

6

Sym(3) × C3

6

Sym(3) × C3

Γ6,6

6

Sym(6)

6

Alt(6)

Table 1. Local actions for some small BMW-groups

Notice that the local action of ΓWise on both tree factors is intransitive. Nevertheless ΓWise has non-discrete projections on both Aut(TA ) and Aut(TX ) by [132, Theorem 5.3] and is thus irreducible by Theorem 4.9. This shows that the irreducibility criterion for BMW-groups derived from Corollary 4.13 and Theorem 4.9 provides a sufficient condition which is not necessary. However, that condition can be used to (re)check the irreducibility of the lattices Γ3,3 , ΓRung , ΓSV , ΓJW , Γ4,5 , ΓRatt and Γ6,6 . 4.6

Residual finiteness

In a Hausdorff topological group, the centralizer of any subset is closed. In particular, in a residually finite group, the centralizer of any subset is separable. This observation yields the following basic fact, where G(∞) denotes the finite residual of G, as in the proof of Proposition 1.1. Lemma 4.14 Let G be a group. For any H ≤ G we have [CG (H), H] ⊂ G(∞) . That observation was used by D. Wise in [130, 132] in order to construct BMWgroups that are not residually finite in the following way. If G is a group and H ≤ G is an inseparable subgroup, then the double of G over H, defined as the free amalgamated product G ∗H G of two copies of G, is not residually finite,

Caprace: Finite and infinite quotients of discrete and indiscrete groups

39

since it admits an involutory automorphism swapping the two factors, and whose centralizer coincides with H. Combining that idea with Proposition 4.10, we obtain the following. Proposition 4.15 (D. Wise) Let Γ = A ∪ X | R be a BMW-presentation of an irreducible BMW-group of degree (m, n). Let A → A¯ : a → a ¯ be a bijection between ¯ and set A and a set A, ¯ 2 = {¯ R a2 | a ∈ A, a2 ∈ R}, ¯ 4 = {¯ ax¯ a x | a, a ∈ A ∪ A−1 , x, x ∈ X ∪ X −1 , axa x ∈ R} R and ¯=R ¯2 ∪ R ¯4. R ¯ ∼ Then Λ = A ∪ A¯ ∪ X | R ∪ R = Γ ∗X Γ is a BMW-presentation of an irreducible BMW-group of degree (2m, n) which is not residually finite. More precisely, there exist a = b ∈ A such that ab−1¯b¯ a−1 lies in the intersection of all finite index subgroups of Λ. Applying that result to ΓWise , D. Wise [130] obtained a non-residually finite BMW-group of degree (8, 6), which was also the first example of a finitely presented non-residually finite small cancellation group and thereby answered negatively a question of P. Schupp [113] from 1973. M. Burger and S. Mozes also constructed non-residually finite BMW-groups using another closely related method that they developed independently. The following result, which illustrates their method, was proved by them in the special case where the local action of Γ on both trees TA and TX is quasi-primitive, see [22, §2.1, 2.2, 2.3 and 6.1]. The proof of the present version will be discussed below. Proposition 4.16 Let Γ = A ∪ X | R be a BMW-presentation of an irreducible BMW-group of degree (M, N ) = (2m + m , 2n + n ), where the notation is as in Proposition 4.2. Let



 ˜ 2 = (a1 , a2 )2 | a1 , a2 ∈ A, a21 , a22 ∈ R ∪ (x1 , x2 )2 | x1 , x2 ∈ X, x21 , x22 ∈ R R and ˜4 = R

         (a1 , a2 )(x1 , x2 )(a1 , a2 )(x1 , x2 ) | a1 x1 a1 x1 ∈ R, a2 x2 a2 x2 ∈ R .

Then ˜2 ∪ R ˜4 Γ  Γ = (A × A) ∪ (X × X) | R 2

is an irreducible BMW-group of degree 2m + 4mm + (m )2 , 2n2 + 4nn + (n )2 which is not residually finite.

40

Caprace: Finite and infinite quotients of discrete and indiscrete groups

The criterion developed by Burger–Mozes in order to prove their version of Proposition 4.16 ensures that, under suitable conditions on the local action, an irreducible lattice in Aut(T1 ) × Aut(T2 ) whose projections to at least one of the factors is not injective, cannot be residually finite, see [22, Proposition 2.1]. That criterion was subsequently generalized to lattices in products of CAT(0) spaces [32, Proposition 2.4], and then to irreducible lattices in products of locally compact groups in [29, Corollary 33] and [34, §5]. We record the following geometric version, where there is no condition on the local action. Proposition 4.17 (Caprace–Monod [32, Proposition 2.4]) Let n ≥ 2 and for each i = 1, . . . , n, let Ti be a locally finite tree with infinitely many ends and no vertex of degree 1. Let also Γ ≤ Aut(T1 ) × · · · × Aut(Tn ) be a discrete subgroup acting cocompactly on T1 × · · · × Tn  . Assume that for each nonempty proper subset I  {1, . . . , n}, the projection Γ → i∈I Aut(Ti ) has non-discrete image. If there exists a nonempty proper subset I  {1, . . . , n} such that the projection  Γ → i∈I Aut(Ti ) fails to be injective, then Γ is not residually finite. This rather general criterion applies to numerous examples. First of all, Propositions 4.15 and 4.16 can both be derived from it. Consider for example the set-up of Proposition 4.15. Let π : Λ = Γ∗X Γ → Γ be the natural homomorphism, which is injective on X . Observe that H = X is a commensurated subgroup of Λ, i.e., for every g ∈ Λ, we have [H : H ∩ gHg −1 ] < ∞. In particular, given g ∈ Λ and h ∈ H ∩ g −1 Hg, we have ghg −1 = h ∈ H. If now g ∈ Ker(π) and recalling that π is the identity on H, we see that h = π(h ) = π(ghg −1 ) = π(h) = h. Thus any g ∈ Ker(π) centralizes a finite index subgroup of H = X , namely H ∩ g −1 Hg. Therefore Ker(π) acts trivially on the tree TX , which is the Cayley graph of X by Proposition 4.2. Thus Ker(π), which contains all elements of the form a¯ a−1 with a ∈ A, is contained in the kernel of the projection of Λ to Aut(TX ). Proposition 4.17 ensures that Λ is not residually finite. The proof of Proposition 4.16 follows by similar considerations. An alternative approach providing also explicit elements in Λ(∞) is as follows. Using Proposition 4.10 and applying Lemma 4.14 to the extension of Λ by the automorphism of order 2 fixing X pointwise and mapping a to a ¯ for all a ∈ A, we see that there exist a = b ∈ A such that ab−1¯b¯ a−1 ∈ Λ(∞) . Proposition 4.17 also applies to other situations. For instance, it implies that the group ΓBDR is not residually finite: Indeed, the projection of the free group a, b, c on the automorphism group of the tree T{x,y,z} is not injective, since its image is isomorphic to a lamplighter group by Proposition 4.6. Similarly, a computation shows that in the group ΓJW from Example 4.3, the elements x3 and y 3 each centralize a subgroup of index 4 of the free group a, b . It follows that x3 , y 3 lies in the kernel of the action of ΓJW on TA = T{a,b} . Hence ΓJW is not residually finite. One can show in a similar way that the Wise lattice is not residually finite. The non-residual finiteness of ΓJW and ΓWise was recently observed independently in [16] and [34]. The groups ΓBDR , ΓJW and ΓWise have small degree, and this was essential in the verification that they fulfill the non-injectivity hypothesis of Proposition 4.17. The

Caprace: Finite and infinite quotients of discrete and indiscrete groups

41

following result provides a general criterion in terms of local actions that allows one to check the hypothesis of Proposition 4.17 on BMW-groups of arbitrary degree. Proposition 4.18 (Caprace–Wesolek) Let n ≥ 2 and for each i = 1, . . . , n, let Ti be a locally finite tree with infinitely many ends and no vertex of degree 1. Let also Γ ≤ Aut(T1 ) × · · · × Aut(Tn ) be a discrete subgroup acting cocompactly on T1 × · · · × Tn . Then the following assertions hold for any i ∈ {1, . . . , n}. and the local action of Γ in Ti is (i) If the Γ-action on Ti is vertex-transitive,  nilpotent, then the projection Γ → j =i Aut(Tj ) is not injective.  (ii) If in addition the projection Γ → j =i Aut(Tj ) has a non-discrete image, then Γ is not residually finite. Proof Assertion (i) follows from [34, Corollary 5.3, Lemma 5.10 and Proposition 6.3]. Assertion (ii) follows from [34, Corollary 6.4].  The following consequence is immediate (see Table 1). Corollary 4.19 Let Γ = A ∪ X | R be a BMW-presentation. If Γ is irreducible and if the local action of Γ on TA or on TX is nilpotent, then Γ is not residually finite. In particular, the BMW-groups ΓJW , ΓWise and Γ3,3 are not residually finite. Remark 4.20 An explicit non-trivial element of the finite residual of ΓJW can be obtained as follows. We mentioned above that in the group ΓJW , the elements x3 and y 3 each commute with a subgroup of index 4 of A , where A = {a, b}. More precisely x3 centralizes the stabilizer A x ≤ A of the edge labelled by x emanating from 1X in the tree TX , while y 3 centralizes the stabilizer A y ≤ A of the edge labelled by y. Moreover the groups A x and A y are both contained in A∗2 , which is a subgroup of index 2 in A . On the other hand, Proposition 4.10 ensures that the profinite closure A∗2 contains a non-trivial element of X ∗2 . Analyzing the local action of ΓJW on TX , one checks that this element is x2 or y 2 . In view of the presentation of ΓJW , the assignments a → a−1 , b → b−1 , x → y −1 and y → x−1 extend to an automomorphism of ΓJW . It follows that the profinite closure A∗2 contains both x2 and y 2 . Since A x and A y are both index 2 subgroups of A∗2 , (∞) we have x4 , y 4 ∈ A x ∩ A y . By Lemma 4.14, we have [x3 , A x ] ⊂ ΓJW and (∞) [y 3 , A y ] ⊂ ΓJW . We deduce that the commutators [x3 , y 4 ] and [y 3 , x4 ] are both (∞) non-trivial elements of ΓJW . 4.7

The Normal Subgroup Theorem, after U. Bader and Y. Shalom

We have already seen examples of BMW-groups that are hereditarily just-infinite, see Propositions 4.4, 4.5 and 4.7. Those groups are irreducible arithmetic lattices in products of two simple algebraic groups of rank 1 (in characteristic 3, 0 and 2 respectively), and the just-infinite property was established using the Margulis Normal Subgroup Theorem. A remarkable breakthrough due to M. Burger and

42

Caprace: Finite and infinite quotients of discrete and indiscrete groups

S. Mozes [22, Theorem 4.1] was to extend the scope of the Margulis Normal Subgroup Theorem so that it applies to a much larger class of BMW-groups, including non-arithmetic (and even non-linear) ones. The Burger–Mozes Normal Subgroup Theorem applies to irreducible cocompact lattices in products of certain virtually simple locally compact groups of automorphisms of trees. It was generalized later by U. Bader and Y. Shalom [7], who obtained a fundamental result whose level of generality is absolutely stunning. In order to present it, we first recall that a locally compact group is called just-non-compact if it is not compact and if all its proper Hausdorff quotients are compact. It is hereditarily just-non-compact if every finite index open subgroup is just-non-compact. The main example of a hereditarily just-non-compact group to keep in mind is that of a topologically simple group, i.e., a non-trivial locally compact group whose only closed normal subgroups are the trivial ones. An example of a just-non-compact locally compact group which is not hereditarily so is provided by the wreath product S  C2 of a topologically simple group S with a cyclic group of order 2. A general result on the structure of a compactly generated just-non-compact locally compact group can be found in [31, Theorem E] (see also [30, Proposition 2.7]). Theorem 4.21 (U. Bader and Y. Shalom) Let n ≥ 2 and for each i = 1, . . . , n, let Gi be a non-discrete compactly generated locally compact group which is just-non-compact and contains no abelian normal subgroup other than the identity subgroup. Let Γ< G1 × · · · × Gn be a cocompact lattice such that for all j, the projection Γ → i =j Gi has dense image. Then Γ is just-infinite. If in addition Gi is hereditarily just-non-compact for all i, then Γ is hereditarily just-infinite. Proof Using [31, Theorem E] and [30, Lemma 4.10], we see that for all j, the projection pj : Γ → Gj is injective. Let now N = {1} be a normal subgroup of Γ. Then for all j, the group pj (N ) is a non-trivial closed normal subgroup of Gj . It is thus cocompact in Gj since the latter is just-non-compact. It then follows from [7, Theorem 3.7(iv)] and [116, Theorem 0.1] that the quotient Γ/N is finite. Hence Γ is just-infinite. Assume now that Gi is hereditarily just-non-compact for all i, and let Λ ≤ Γ be a finite index subgroup of Γ. We may then replace Gi by pi (Λ) for all i and apply the first part of the proof. This shows that Λ is just-infinite. Hence Γ is hereditarily just-infinite.  The condition excluding abelian normal subgroups in Gi in Theorem 4.21 is necessary: it is easy to see that the isometry group G = Isom(R) is just-noncompact, and that the discrete group Γ = (Z × Z)  C2 , where the generator of C2 acts via (a, b) → (−a, −b), is not just-non-compact, but embeds as a cocompact lattice with dense projections in G × G. Similarly, the condition of compact generation is also necessary. In order to see that, consider the group G1 = PSL2 (Fp )  Aut(Fp ), where Fp denotes the algebraic closure of the finite field of order p. The group G1 carries a second countable locally compact group topology which induces the discrete topology on

Caprace: Finite and infinite quotients of discrete and indiscrete groups

43

the countable subgroup PSL2 (Fp ), and the Krull topology (which is compact) on the Galois group Aut(Fp ). Since the discrete group PSL2 (Fp ) is locally finite, it follows that every compactly generated closed subgroup of G1 is compact. Since G1 is not compact, it follows that it cannot be compactly generated. On the other hand, it is not difficult to show that every non-trivial normal subgroup of G1 contains PSL2 (Fp ), so that G1 is just-non-compact (and hereditarily so). We also set G2 = Z and Γ = PSL2 (Fp ) α Z, where α is a generator of the pro-cyclic group Aut(Fp ). The group Γ embeds as a cocompact lattice with dense projections in G1 × G2 . Both factors in that product are hereditarily just-non-compact, but Γ maps onto Z and thus fails to be just-infinite. Theorem 4.21 also holds for some non-uniform lattices under a technical hypothesis called integrability, see [7]. This was exploited in [33] to exhibit another family of finitely presented infinite simple groups coming from Kac–Moody theory; we will not pursue that direction here. The proof of Theorem 4.21 follows the scheme designed by Margulis in the proof of his own Normal Subgroup Theorem. The required conclusion is obtained by combining two independent results, proved separately with completely different methods: the first ensures that every proper quotient of Γ has Kazhdan’s property (T), the second ensures that every proper quotient of Γ is amenable (see [10] for a detailed exposition of those important notions). The conclusion follows since the only discrete groups that satisfy both property (T) and amenability are finite. In fact, some parts of Margulis’ original proof were already formulated for a rather general class of locally compact groups without any hypothesis requiring that those are algebraic over local fields, see [83, Theorem 1.3.2]. The property (T) half of the scheme above was achieved at the greatest level of generality by Y. Shalom [116], while the amenability half is the work of Bader–Shalom [7]. 4.8

Alternating and fully symmetric local actions

In order to apply the Normal Subgroup Theorem 4.21 to a BMW-group given by its BMW-presentation, one should check that the closure of its projections on the automorphism groups of both tree factors are just-non-compact. Once again, there is no general tool allowing one to describe the closure of a non-discrete subgroup of the automorphism group of a tree or any other connected locally finite graph. In addressing that issue, M. Burger and S. Mozes highlighted a very striking phenomenon, allowing them to control the closure of a non-discrete vertex-transitive automorphism group of a tree under the hypothesis that the local action has simple (or almost simple) point stabilizers. In order to give a precise statement, we need to recall the definition of the Burger–Mozes universal group of automorphisms of the d-regular tree T with local action prescribed by a permutation group F ≤ Sym(d), introduced in [21, §3.2]. Fix a map i : E(T ) → {1, . . . , d} such that for every vertex v ∈ V (T ), the restriction i|E(v) of i to the set of edges containing v is a bijection. Given g ∈ Aut(T ) and v ∈ V (T ), we set σ(g, v) = i|E(gv) ◦ g ◦ (i|E(v) )−1 .

44

Caprace: Finite and infinite quotients of discrete and indiscrete groups

Notice that σ(g, v) ∈ Sym(d) for all g and v. Given any F ≤ Sym(d), the Burger– Mozes universal group of automorphisms of T with local action prescribed by F is defined as U (F ) = {g ∈ Aut(T ) | σ(g, v) ∈ F for all v ∈ V (T )}. One checks that, up to conjugation in Aut(T ), it is independent of choice of the map i. If d ≥ 3 and if the group F is transitive and generated by its point stabilizers, then the index 2 subgroup U (F )+ of U (F ) preserving the canonical bipartition of T is a simple compactly generated locally compact group, see [21, Proposition 3.2.1]. Theorem 4.22 (Burger–Mozes [21, Propositions 3.1.2, 3.3.1 and 3.3.2]) Let T be a d-regular tree with d ≥ 6 and Γ ≤ Aut(T ) be a non-discrete vertextransitive subgroup. (i) If the local action of Γ in Aut(T ) contains the full alternating group Alt(d), then the closure Γ is hereditarily just-non-compact, without abelian normal subgroup other than {1}. (ii) If the local action of Γ in Aut(T ) coincides with the full alternating group Alt(d), then the closure Γ is conjugate to the universal group U (Alt(d)) in Aut(T ). In particular Γ has a simple subgroup of index 2. One should keep in mind that, in view of Corollary 4.13, the non-discreteness of Γ can be checked in a ball of radius 3 under the assumption that the local action contains the alternating group of degree ≥ 6. The uniqueness (up to conjugacy) of the non-discrete vertex-transitive closed subgroup of Aut(T ) established in Theorem 4.22(ii) is rather surprising. The complete classification of the non-discrete vertex-transitive closed subgroups of Aut(T ) whose local action is the full symmetric group was recently achieved by N. Radu. In particular, the following result of his is an important refinement of Theorem 4.22(i). Theorem 4.23 (N. Radu [100, Theorem B and Corollary E]) Let T be a d-regular tree with d ≥ 6 and Γ ≤ Aut(T ) be a non-discrete vertex-transitive subgroup. If the local action of Γ in Aut(T ) contains the full alternating group Alt(d), then the closure Γ has a simple subgroup of index ≤ 8, and belongs to an explicit infinite list of examples. We refer to [100] for a description of those examples, and for a more general classification result that does not require any hypothesis of vertex-transitivity. We emphasize the contrast between Theorem 4.22(ii) and Theorem 4.23: if the local action is Alt(d), then the non-discrete group Γ is uniquely determined, whereas if the local action is Sym(d), there are infinitely many pairwise non-conjugate possibilities for Γ.

Caprace: Finite and infinite quotients of discrete and indiscrete groups 4.9

45

Virtually simple BMW-groups of small degree

By way of illustration, let us consider the BMW-group Γ6,6 introduced in Proposition 4.8. It is irreducible of degree (6, 6), and in view of Table 1, its local action on the two tree factors TA and TX is Sym(6) and Alt(6) respectively. In view of Theorems 4.21 and 4.22, it follows that Γ6,6 is hereditarily just-infinite. From Theorem 4.22 we deduce moreoveor that the closure of the projection of Γ to Aut(TX ) is isomorphic to U (Alt(6)). The closure of the projection of Γ to Aut(TA ) has also been identified by N. Radu [101]: it is isomorphic to the subgroup G(i) ({1}, {1}) ≤ Aut(TA ) in the notation of [100, Definition 4.9]. It has a simple subgroup of index 4. Since Γ6,6 also contains the non-residually finite group ΓJW as a subgroup (see Proposition 4.8), it follows from Proposition 1.1 that Γ6,6 is virtually simple. In fact, the following more precise assertion holds. Proposition 4.24 (N. Radu [99]) The finite residual of BMW-group Γ6,6 coin+ cides with its subgroup Γ+ 6,6 of index 4. In particular Γ6,6 is a finitely presented torsion-free simple group which splits as an amalgamated free product of the form F5 ∗F25 F5 . Similarly, the index 4 subgroup Γ+ 4,5 ≤ Γ4,5 is simple, and splits as an amalgamated free product of the form F3 ∗F11 F3 . (∞)

Proof By Remark 4.20, we have [x3 , y 4 ] ∈ ΓJW . Moreover ΓJW is a subgroup of Γ6,6 by Proposition 4.2, so that the finite residual of Γ6,6 also contains [x3 , y 4 ]. Since Γ6,6 is hereditarily just-infinite and non-residually finite, its finite residual is simple and coincides with its smallest non-trivial normal subgroup, see Proposition 1.1. In particular it coincides with the normal closure of [x3 , y 4 ] in Γ6,6 . Using a computer algebra software like GAP, it takes a couple of seconds to check that the quotient of the just-infinite finitely presented group Γ6,6 by the normal closure of [x3 , y 4 ] is of order 4. This completes the proof in the case of Γ6,6 in view of Proposition 4.2. The proof for Γ4,5 follows a similar outline. However, additional arguments are required to check that the hypotheses of Theorem 4.21 are satisfied: indeed, the degree is too small for Theorem 4.22 to apply. We refer to [99, Theorem 5.5 and Corollary 5.6] for the details.  The original paper of Burger–Mozes [22] provided the first examples of virtually simple BMW-groups. The degrees of those examples are rather large, and the finite index simple subgroup of the smallest example decomposes as an amalgamated free product of the form F217 ∗F75601 F217 . The reason why the degree of those examples is large is that Burger–Mozes relied on Proposition 4.16 applied to an arithmetic BMW-group Γ to build a non-residually finite BMW-group. The smallest example produced in that way already has a rather large degree. The degree was then increased in order to build an example of a BMW-group containing that nonresidually finite one, and which moreover has alternating local actions on both tree factors, and is thus also hereditarily just-infinite by Theorems 4.21 and 4.22. Much smaller examples were constructed by D. Rattaggi [106, 108] using the nonresidual finiteness of the double of the Wise lattice. In that way, he obtained

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a torsion-free simple group which decomposes as F7 ∗F73 F7 . Exploiting the fact that the Wise lattice itself is non-residually finite, an example decomposing as F7 ∗F49 F7 is constructed in [16]. We refer to [101] for a list of over a hundred other examples of BMW-groups of degree (4, 5) and (6, 6) similar to Γ4,5 and Γ6,6 , that are virtually isomorphic to a simple group of the form F3 ∗F11 F3 or F5 ∗F25 F5 . By [99, Corollary IV], the group Γ+ 4,5 admits the following presentation, which witnesses its amalgam decomposition as F3 ∗F11 F3 : ∼ Γ+ 4,5 = a, b, c, x, y, z | a = x, b2 = yx−1 y, c2 = z 2 , c−1 ac = z −1 yz, c−1 bc = z −1 xz, b−1 ab = y −1 x−1 y, b−1 c−2 b = y −1 xz −2 y, b−1 c−1 b−1 acb = y −1 xz −1 x−1 yzx−1 y, b−1 c−1 abcb = y −1 xz −1 yz −1 xzx−1 y, b−1 c−1 b2 cb = y −1 xz −1 xz −1 xzx−1 y, b−1 c−1 b−1 cbcb = y −1 xz −1 x−1 z −1 xzx−1 y . 4.10

The hyperbolic manifold analogy Nous voyons donc d´ej` a que les analystes ne sont pas de simples faiseurs de syllogismes ` a la fa¸con des scolastiques. Croira-ton, d’autre part, qu’ils ont toujours march´e pas ` a pas sans avoir la vision du but qu’ils voulaient atteindre? Il a bien fallu qu’ils devinassent le chemin qui y conduisait, et pour cela ils ont eu besoin d’un guide. Ce guide, c’est d’abord l’analogie. Henri Poincar´e, La Valeur de la Science, 1905

As we have seen, the proof of existence of virtually simple BMW-groups by Burger–Mozes elaborates on techniques initially developed by Margulis in his seminal study of irreducible lattices in semi-simple groups of rank ≥ 2. In this speculative section, we discuss an analogy between BMW-complexes and real hyperbolic closed manifolds which suggests that BMW-groups are also strongly related to lattices in simple Lie groups of rank 1. That analogy could serve as an invitation for further research. Every rank 1 simple Lie group is locally isomorphic to one of the following groups: • The isometry group of the real hyperbolic n-space, denoted O(n, 1). • The isometry group of the complex hyperbolic n-space, denoted U (n, 1).

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• The isometry group of the quaternionic hyperbolic n-space, denoted Sp(n, 1). For n = 1, that space is isometric to the real hyperbolic 4-space. • The isometry group of the octonionic hyperbolic plane, denoted F4−20 . Fundamental results of K. Corlette [39] and Gromov–Schoen [57] ensure that every lattice in Sp(n, 1) (with n ≥ 2) and in F4−20 is arithmetic. We refer to [85] for the formal definition of arithmeticity. Let us merely mention Margulis’ criterion [85, Theorem IX.1.10] according to which a lattice in a simple Lie group is arithmetic if and only if its commensurator is discrete. The commensurator of a subgroup Γ of a group G is the set CommG (Γ) = {g ∈ G | [Γ : Γ ∩ gΓg −1 ] < ∞, [Γ : Γ ∩ g −1 Γg] < ∞}, which is a subgroup of G. Lattices in O(n, 1) and U (n, 1) can be non-arithmetic. While infinitely many examples of non-arithmetic lattices in O(n, 1) are known for all n ≥ 2, only finitely many non-arithmetic lattices in U (n, 1) are known, see [45] and references therein for the current state of the art. Finding infinite families of non-arithmetic lattices in U (n, 1) is a major challenge. The case of O(n, 1) is much better understood. In particular, it is known that a huge majority of the lattices in O(n, 1) are nonarithmetic. This is made precise by the following result, which combines important contributions of various authors. Theorem 4.25 Let n ≥ 4 and for each v > 0, let Cnc (v) be the number of commensurability classes of real hyperbolic closed manifolds of dimension n admitting a representative of volume ≤ v, which is finite by a classical result of H. C. Wang [123]. Let also Cnarith (v) be the number of commensurability classes of real hyperbolic closed manifolds of dimension n admitting a representative of volume ≤ v whose fundamental group is arithmetic. Then there exist positive constants a, b, c, ε such that the following inequalities hold for all sufficiently large v: (i) (Burger–Gelander–Lubotzky–Mozes [19]) Cnc (v) ≤ v bv . (ii) (Gelander–Levit [48]) Cnc (v) ≥ v av . ε

(iii) (M. Belolipetsky [11]) Cnarith (v) ≤ v c(log v) . In particular, we see that the proportion of non-arithmetic real hyperbolic closed manifolds becomes strikingly overwhelming as the volume tends to infinity. It is conjectured that the upper bounded on Cnarith (v) given in Theorem 4.25(iii) can be improved to a polynomial bound (i.e., ε = 0). This has in fact been proved by M. Belolipetsky for non-compact manifolds; it is open in the compact case. The enumeration of BMW-presentations of small degree in [106] and [101] suggests a similar counting problem. The direct analogue would be to fix the degree (m, n) and count the number of commensurability classes of cocompact lattices in Aut(Tm ) × Aut(Tn ) as a function of the covolume. However, formulating any conjecture about that number is premature, since there is currently no known evidence that the number of those commensurability classes actually grows at all for

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all values of m and n. Instead, we focus on BMW-groups of degree (m, n) (so that the covolume is fixed, once the Haar measure on Aut(Tm ) × Aut(Tn ) is normalized so as to give measure 1 to the vertex stabilizers) and address the counting problem of the number of their commensurability classes as a function of the degree. A BMW-group is called arithmetic if the closure of its projections to the automorphism groups of the tree factors of its Cayley graph are both rank 1 simple algebraic groups over local fields. Problem 4.26 Let BMW(m, n) be the number of commensurability classes of BMW-groups of degree (m, n), and let moreover BMWarith (m, n) be the number of those classes that have an arithmetic representative. Determine the asymptotic growth type of BMW(m, n) and BMWarith (m, n) as functions of (m, n). In particular, determine whether there exist constants a, b, c such that the following inequalities hold for all sufficiently large m, n ∈ N, where w = m + n: (i) waw ≤ BMW(m, n) ≤ wbw . (ii) BMWarith (m, n) ≤ wc . We note that any two finitely generated free groups are commensurable, so that the reducible BMW-complexes only contribute one commensurability class per degree. We also note that the Buhat–Tits tree of a rank 1 simple algebraic group over a local field is always semi-regular of bidegree (q + 1, q  + 1), where q and q  are both powers of the same prime. Thus for most values of (m, n), there is no arithmetic BMW-group of degree (m, n) whatsoever. As pointed out to me by A. Vdovina, the results by Stix–Vdovina [117] ensures that if 2n − 1 is a sufficiently large prime power, then the number of commensurability classes of arithmetic torsion-free BMW-groups of degree (2n, 2n) is bounded below by a linear function of n. The difficulty in controlling the commensurability classes is that the currently known tools do not provide detailed enough information on the finite quotients of a BMW-group. We have pointed out the existence of BMW-groups of degree (4, 5) and (6, 6) that are virtually simple. The proof of non-residual finiteness relied on a very specific property, namely the existence of a BMW-subgroup that is not residually finite. The results of N. Radu [101] show that there are 225145 isomorphism classes of BMW-presentations of degree (6, 6) whose generators have infinite order. Among them, 23225 yield a BMW-group whose local action on both tree factors is isomorphic to Alt(6) or Sym(6); these groups are all hereditarily just-infinite, see Proposition 4.32 below. I expect that they are all virtually simple. However, this has only been checked for 96 of them (and these 96 groups have moreover been proved to be pairwise non-commensurable). Those are precisely the members of that list of 23225 BMW-groups that contain an isomorphic copy of the non-residually finite group ΓJW . The very small size of the ratio 96/23225  0.4% illustrates our lack of understanding: approaching Problem 4.26 requires new and much finer criteria to check non-residual finiteness.

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The following question, which overlaps several problems posed by D. Wise in [132, Section 10], provides further illustrations of the limitations of the currently known tools. Problem 4.27 Is there an algorithm which determines whether a BMW-group given by a BMW-presentation is irreducible? Residually finite? Linear? Arithmetic? Just-infinite? Virtually simple? Has a discrete commensurator in the full automorphism group of the associated product of trees? Is there an algorithm which determines whether two such groups are isomorphic? Commensurable? A complete solution to Problem 4.27 would of course be spectacular, but such a complete result is not necessary to tackle Problem 4.26. Indeed, it could be that the BMW-groups with a locally alternating or fully symmetric action on both tree factors already contribute enough commensurability classes to dominate the growth function from the conjecture. 4.11

Local actions of just-infinite groups acting on trees

We have seen how the combination of Theorems 4.21 and 4.22 (and its companion Theorem 4.23) can be used to build cocompact lattices Γ ≤ Aut(T ) × Aut(T  ) in the automorphism group of a product of two regular locally finite trees of degrees d and d respectively, that are (hereditarily) just-infinite. A specific feature of those examples, coming from the hypotheses of Theorem 4.22, is that the local actions of Γ on both T and T  contain Alt(d) and Alt(d ). We have also seen that other just-infinite lattices in products of trees arise as arithmetic groups (see Propositions 4.4, 4.5 and 4.7); in those examples, the local actions on both tree factors are finite Lie type groups of rank 1 acting on projective lines over finite fields (see Table 1). In all known examples of just-infinite lattices in products of trees, the local action on each tree factor is 2-transitive. This observation suggests the following. Problem 4.28 Let T1 , . . . , Tn be locally finite trees all of whose vertices have degree ≥ 3, and Γ ≤ Aut(T1 ) × · · · × Aut(Tn ) be a discrete subgroup acting cocompactly on T1 × · · · × Tn . Assume that Γ is just-infinite. Must the local action of Γ on Ti be 2-transitive for all i = 1, . . . , n? The condition that Γ be just-infinite implies that n ≥ 2, since a discrete cocompact automorphism group of a single infinitely-ended tree is virtually a non-abelian free group. One could also ask the following more general question. Problem 4.29 Let T be a locally finite tree all of whose vertices have degree ≥ 3, and Γ ≤ Aut(T ) be a (not necessarily discrete) subgroup acting cocompactly. What are the possible local actions of Γ at vertices of T if Γ is finitely generated and just-infinite?

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As an illustration of this problem, let us mention that if Γ is finitely generated and its local action at every vertex of T is nilpotent, then Γ is virtually indicable (see [34, Corollary 1.2]); in particular Γ cannot be just-infinite if T has infinitely many ends. Problem 4.29 is closely related to the statement (2) given without proof in [8, Section 9.15] and attributed to E. Rips, according to which a group Γ of the form A ∗C B with C = B and such that |C\A/C| ≥ 3 must be SQ-universal. In other words, that claim would imply that if Γ is edge-transitive on T and if every vertex of T has valency ≥ 2, then Γ is SQ-universal as soon as its local action at one vertex of valency ≥ 3 fails to be 2-transitive. However, that statement must be amended: indeed, the following result of A. Le Boudec provides a counterexample. Theorem 4.30 (A. Le Boudec) There is a finitely generated infinite simple group Γ acting edge-transitively on the regular tree T of degree 20, and whose local action at every vertex is isomorphic to the permutational wreath product Alt(4)  Alt(5). In particular the local action of Γ at every vertex of T is not primitive. Proof We apply [76, Theorem 1.3] to the group F = C2 × C2 × C5 of order 20, and the group F  = Alt(4)  Alt(5), both viewed as transitive subgroups of Sym(20). Notice that the permutation group F can naturally be viewed as a subgroup of F  . Since F acts freely and F  is generated by the derived subgroups of its point stabilizers, the existence of the required simple group Γ ≤ Aut(T ), arising as an index 2 subgroup of a vertex-transitive group denoted by G(F, F  ), indeed follows from [76, Theorem 1.3].  In particular, a finitely generated infinite simple group can act edge-transitively on a tree with an imprimitive local action at every vertex. Note however that the group Γ from Theorem 4.30 is very different from the projection of a lattice as in Problem 4.28, so that Theorem 4.30 should not be interpreted as evidence supporting a negative solution to Problem 4.28. Indeed, in the set-up of Problem 4.28, the group Γ is finitely presented, and the vertex-stabilizers for the Γ-action on each tree factor Ti are not torsion groups (they act properly and cocompactly on the product of the tree factors different from Ti ). On the other hand, Le Boudec’s group has locally finite vertex stabililizers (see the discussion at the end of Section 3.1 in [76]); moreover it is not finitely presented (see [76, Proposition 5.4]). 4.12

Lattices in products of more than two trees

We close this chapter with a discussion of another fascinating and natural problem. Products of two trees are part of the definition of a BMW-group, but it is natural to consider also products of more than two factors. In view of the existence of virtually simple BMW-groups, the following problem is especially intriguing. Problem 4.31 Let T1 , . . . , Tn be locally finite trees with infinitely many ends and Γ ≤ Aut(T1 ) × · · · × Aut(Tn ) be a discrete subgroup acting cocompactly on T1 × · · · × Tn . Can Γ be simple if n ≥ 3?

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This problem may be viewed as a higher rank version of P. Neumann’s Question 2.7. The arithmetic constructions in Section 3.2 provide examples of cocompact lattices in products of an arbitrarily large number of factors that are hereditarily just-infinite. Explicit examples acting vertex-transitively on the associated product of trees may be found in [37, Corollary 6.2] and [36]. All those groups are linear, hence residually finite. There is currently no known example of a lattice in a product of more than two non-linear locally compact groups satisfying the hypotheses of the Bader–Shalom Normal Subgroup Theorem 4.21. A non-existence result for irreducible lattices in products of three or more factors has been established in the restricted class of certain locally compact Kac–Moody groups in [32]. The techniques used in loc. cit. are very specific to Kac–Moody theory, and yield no relevant information in the context of Problem 4.31. In the case of tree lattices, the following result was established by N. Radu with the aid of a computer. Proposition 4.32 (N. Radu [99, Theorem VIII]) copies of the regular tree of degree 6.

Let T , T  , T  be three

(i) There are 23225 conjugacy classes of subgroups Γ ≤ Aut(T ) × Aut(T  ) acting simply transitively on the vertices of the product T × T  , whose local action on both tree factors is Alt(6) or Sym(6), and such that Γv and Γv are torsion-free for all v ∈ V (T ) and v  ∈ V (T  ). All of them are hereditarily just-infinite. Among them, 2240 are torsion-free. (ii) There is no subgroup Γ ≤ Aut(T ) × Aut(T  ) × Aut(T  ) acting simply transitively on the vertices of the product T × T  × T  , such that the following conditions hold, where G, G and G denote the closure of the respective projections of Γ to Aut(T ), Aut(T  ) and Aut(T  ): • the groups G, G and G are non-discrete and their respective local actions at every vertex of T , T  and T  is Alt(6) or Sym(6); • Γv,v and Γv ,v are torsion-free for all (v, v  , v  ) ∈ V (T × T  × T  ). Part (i) of Proposition 4.32 relies on a version of Mostow rigidity for irreducible cocompact lattices in products of trees with primitive local action, due to Burger– Mozes–Zimmer, see [23, Theorem 1.4.1]. Part (ii) relies in an essential way on (i), see [99, Theorem VIII]. In particular, it uses the Normal Subgroup Theorem in the case of two factors. Notice that the condition on the local action hypothetized in Proposition 4.32 ensures that G, G and G are subjected to Theorem 4.23. That condition is rather natural, especially if one expects a positive solution to Problem 4.28 (bearing in mind that ‘almost all’ finite 2-transitive groups are Alt(d) or Sym(d); see [100, Corollary B.2] for a precise statement clarifying the latter claim). The contrast between the case of 2 and 3 factors in Proposition 4.32 is striking. I interpret it as experimental evidence for a negative answer to Problem 4.31.

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Quotients of hyperbolic groups and asymptotic properties of finite simple groups

In this final chapter, the difference between the construction of finite and infinite quotients of finitely presented groups is further illustrated by the discussion of a major open problem in Geometric Group Theory, namely the residual finiteness of hyperbolic groups. 5.1

Examples of hyperbolic groups

Hyperbolic groups form a class of groups introduced and developed by M. Gromov in [56]. Their definition can be seen as an axiomatization of the fundamental groups of hyperbolic closed manifolds. Extensive treatments of the basic theory can be consulted in [17, Chapter III.H], [38] or [49]. Let us record a (non-exhaustive!) list of examples. • Finite groups, and more generally virtually cyclic groups. Those form the so-called elementary hyperbolic groups. • Virtually free groups. Those include free amalgamated products of finite groups, e.g., the free product Ca ∗ Cb of cyclic groups of order a and b. • Fundamental groups of closed surfaces of genus g ≥ 2. • Hyperbolic triangle groups. Those are groups of the form T (p, q, r) = x, y | xp , y q , (xy)r with p1 + 1q + 1r < 1. They are commensurable with surface groups. • One-relator groups with torsion. • Coxeter groups which do not contain Z × Z. The latter condition can be characterized in terms of the Coxeter presentation, see [42, §12.6]. • Fundamental groups of closed Riemannian manifolds of negative sectional curvature. 5.2

Finite and infinite quotients of hyperbolic groups

Hyperbolic groups enjoy numerous remarkable algebraic properties. Let us collect a few of those. Theorem 5.1 Every hyperbolic group G satisfies the following. (i) ([56, Corollary 2.2.A]) G is finitely presented. (ii) ([49, Chapter 8, Theorem 37]) Every subgroup of G is either virtually cyclic, or contains a non-abelian free subgroup. (iii) (T. Delzant [44, Theorem 3.5]; A. Olshanskii [94]) If G is non-elementary, then G is SQ-universal. The SQ-universality of non-elementary hyperbolic groups is a vast generalization of the classical theorem of Higman–Neumann–Neumann [62] according to which

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the free group F2 is SQ-universal. It was announced by M. Gromov [56, §5.6.E] (without proof). It underlines the huge supply of infinite proper quotients that every non-elementary hyperbolic group has. Since the early days of the theory, the question of existence of finite proper quotients arose naturally. The following question is a major open problem in Geometric Group Theory. Problem 5.2 Are all hyperbolic groups residually finite? In his foundational paper on hyperbolic groups, M. Gromov suggested that the answer should be negative by writing the following. Remark 5.3 (M. Gromov [56, §5.3.B]) Probably, “generic” word hyperbolic groups admit no sequences of subgroups of finite index with trivial intersection. Gromov’s remark has sometimes been interpreted as a conjecture predicting the existence of a non-elementary hyperbolic group whose only finite quotient is the trivial one (see for example Olshanskii’s comment following Theorem 2 in [95]). While the latter conjecture clearly implies a negative answer to Problem 5.2, it turns out that, conversely, the existence of a non-residually finite hyperbolic group would imply that the conjecture is true. This was observed independently by Kapovich–Wise [67] and A. Olshanskii [95]. Problem 5.2 has motivated a tremendous amount of research; it goes beyond the scope of this article to survey it all. We will only emphasize specific results that we find most relevant to the general theme of these notes. The following statement illustrates the strength and scope that a positive solution to Problem 5.2 would have. Theorem 5.4 If all hyperbolic groups are residually finite, then the following assertions hold. (i) (Kapovich–Wise [67, Theorem 5.1]) Every hyperbolic group is virtually torsion-free. (ii) (Agol–Groves–Manning [2]) Every quasi-convex subgroup of a every hyperbolic group is separable. (iii) (A. Lubotzky [82, Remark 4.2]) Cocompact lattices in Sp(n, 1) (with n ≥ 2) and F4−20 do not satisfy the Congruence Subgroup Property. A quasi-convex subgroup of a hyperbolic group G is a finitely generated subgroup H such that the inclusion of H in G is a quasi-isometric embedding. In particular, quasi-convex subgroups are themselves hyperbolic. One may interpret Theorem 5.4(ii) as follows: if all hyperbolic groups are residually finite, then all hyperbolic groups have a tremendous amount of finite quotients. This assertion can be made more precise using the following. Proposition 5.5 Let G be a group and H be a subgroup. If every finite index subgroup of H is separable in G, then every homomorphism ϕ : H → Q to a finite

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˜ → Q defined on a finite index subgroup G ˜ group extends to a homomorphism ϕ˜ : G of G containing H. In particular, if G is a non-elementary hyperbolic group all of whose quasi-convex subgroups are separable, then every finite group is a quotient of a finite index subgroup of G. Proof The first assertion follows from the proof of [81, Theorem 4.0.7]. The second follows from the first together with the fact that every non-elementary hyperbolic group admits quasi-convex subgroups that are free of arbitrarily large rank.  One way to establish Assertion (iii) in Theorem 5.4 is to deduce it from Assertion (ii) and Proposition 5.5. Indeed, the property that all finite groups appear as virtual quotients is incompatible with the congruence subgroup property. We conclude this section by mentioning that a far-reaching generalization of Theorem 5.1(iii) on the SQ-universality of non-elementary hyperbolic groups was recently established by Dahmani–Guirardel–Osin [40], who indeed showed that all the so-called acylindrically hyperbolic groups are SQ-universal. Acylindrically hyperbolic groups form an immensely vast class of groups, formalized by D. Osin [96]. They include all non-elementary hyperbolic groups, as well as numerous other examples of a very different nature, see [96, §8]. For example, it is proved in [87, Corollary 4.26] that the Higman group from Theorem 2.2 is acylindrically hyperbolic. Another illustration of this concept is provided by the following result, giving a rather general perspective on the fact that the Margulis’ Normal Subgroup Theorem fails in the rank 1 case. Theorem 5.6 Let G be a locally compact group which is Gromov hyperbolic with respect to the word metric associated with a compact generating set. Assume that G does not contain a discrete cocompact cyclic subgroup. Then every lattice Γ ≤ G is acylindrically hyperbolic, hence SQ-universal. Proof By [27, Proposition 2.1], the group G has a continuous proper cocompact action on a Gromov hyperbolic proper geodesic metric space X. Any discrete subgroup of G thus acts properly on X. Now, if a discrete subgroup Γ ≤ G does not contain any loxodromic element, then by [28, Proposition 5.5] either Γ has a bounded orbit on X, or Γ has a unique fixed point in the Gromov boundary ∂X. If Γ is a lattice, then the former condition would imply that X carries a G-invariant probability measure, hence that G has a bounded orbit (since a sufficiently large ball in X would have measure > 12 ), while the latter would imply that the Gromov boundary ∂X carries a G-invariant probability measure, which is incompatible with the contracting dynamics of the G-action by the hypothesis that G is nonelementary. Thus every lattice in G acts properly on X and contains a loxodromic element. The required conclusions now follow from [96, Theorems 1.2 and 8.1].  5.3

Hyperbolic quotients of hyperbolic groups, after A. Olshanskii

Our next goal is to emphasize the relation between Problem 5.2 and the asymptotic properties of finite simple groups. A relation of that kind was first highlighted by Ivanov–Olshanskii [64, Problem 2].

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Let us first recall from [56, Corollary 8.2.C] that every infinite order element g of a hyperbolic group G is contained in a unique maximal elementary quasi-convex subgroup EG (g), called the elementary closure of g. Given a subgroup H ≤ G, we define EG (H) as the intersection of EG (h) where h runs over the set of all infinite order elements of H. By [93, Proposition 1], if H is not an elementary quasi-convex subgroup of G, then EG (H) coincides with the largest finite subgroup of G normalized by H. Theorem 5.7 (A. Olshanskii [93, Theorem 2] and [95, Lemma 5.1]) Let G be a non-elementary hyperbolic group and H1 , . . . , Hk be non-elementary subgroups with EG (G) = EG (H1 ) = · · · = EG (Hk ) = {1}. Then there is a homomor˜ onto a non-elementary hyperbolic group G ˜ such that ϕ(Hi ) = G ˜ phism ϕ : G → G for all i. Given two non-elementary hyperbolic groups G1 , G2 , let Hi = Gi /EGi (Gi ) and form the free product G = H1 ∗ H2 , which is hyperbolic. Applying Theorem 5.7 to G, we obtain the following. Corollary 5.8 (A. Olshanskii) Any two non-elementary hyperbolic groups have a common non-elementary hyperbolic quotient. A first connection with finite simple groups arises in the following. Corollary 5.9 If all hyperbolic groups are residually finite, then for every nonelementary hyperbolic group G and every n ≥ 5, the group G has a finite simple quotient containing a copy of Alt(n). Proof We use Corollary 5.8 to construct a common hyperbolic quotient Q of G and the virtually free group Alt(n) ∗ Alt(n), which is perfect. If Q has a non-trivial finite quotient, then a smallest such quotient is a finite simple quotient of G which contains Alt(n) as a subgroup.  One may now contemplate again the list of examples of hyperbolic groups mentioned above and wonder whether it contains candidates of groups that do not map onto non-abelian finite simple groups of arbitrarily large rank. We emphasize that the question of determining the finite simple quotients of the virtually free group Ca ∗ Cb or of the hyperbolic triangle group T (p, q, r) are important topics of current investigations, see [70], [77] and references therein. Combining Theorem 5.4 and Proposition 5.5, we see that if all hyperbolic groups were residually finite, then every finite group would be a quotient of a finite index subgroup of every hyperbolic group. The following immediate consequence of Corollary 5.9 strengthens that fact. Corollary 5.10 If all hyperbolic groups are residually finite, then every finite group embeds in some finite quotient of every non-elementary hyperbolic group.

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That property can be viewed as a finite counterpart of the SQ-universality of hyperbolic groups. Let us finally record a last important consequence of Theorem 5.7. Corollary 5.11 If all hyperbolic groups are residually finite, then for every nonelementary hyperbolic group G with EG (G) = {1}, every finite subset M ⊂ G and every n ≥ 5, there is a homomorphism ρ : G → Q of G onto a finite simple group S containing a subgroup isomorphic to Alt(n), and such that the restriction of ρ to M is injective. Proof For every pair x = y ∈ M , let Hx,y denote the normal closure of xy −1 in G. Since EG (G) = {1}, it follows that Hx,y is a non-elementary normal subgroup of G, which moreover satisfies EG (Hx,y ) = {1}. By Theorem 5.7, the hyperbolic group G ∗ Alt(n) ∗ Alt(n) admits a non-elementary hyperbolic quotient ϕ : G ∗ Alt(n) ∗ Alt(n) → Q such that ϕ(Hx,y ) = Q for all x = y ∈ M and moreover ϕ(Alt(n) ∗ Alt(n)) = Q. Assume that Q is residually finite, and let ψ : Q → S be a smallest non-trivial finite quotient of Q. Since Q is generated by two copies of Alt(n), it is perfect and so S is a finite simple group containing Alt(n). Let now x = y ∈ M . Since Hx,y is the normal closure of xy −1 in G and ϕ is surjective, it follows that ϕ(Hx,y ) = Q is the normal closure of ϕ(xy −1 ) in Q. In particular ρ(Hx,y ) = S is the normal closure of ρ(xy −1 ) in S. Therefore ρ(x) = ρ(y).  We recall that if P is a group property (e.g., being finite, or nilpotent, or solvable), a group G is called residually P if every non-trivial element of G remains non-trivial in some quotient of G satisfying P . Moreover G is called fully residually P if for every finite subset M of G, there is a quotient map ρ : G → Q onto some group satisfying P , which is moreover injective on M . With that terminology at hand, we deduce from Corollary 5.11 that if all hyperbolic groups are residually finite, then every non-elementary hyperbolic group G with EG (G) = {1} is fully residually finite simple. The following neat observation was pointed out to me by P. Neumann. Proposition 5.12 (P. Neumann) Let G be a group, all of whose non-trivial normal subgroups have a trivial centralizer. For any group property P , the group G is residually P if and only if G is fully residually P . In particular, a non-elementary hyperbolic group G with EG (G) = {1} or, more generally, an acylindrically hyperbolic group with trivial amenable radical, is residually P if and only if it is fully residually P . Proof Clearly, if G is fully residually P then it is residually P . In order to prove the converse, it suffices to show that for any non-empty finite subset M ⊂ G, there is a quotient map ρ : G → Q onto some group satisfying P , such that ρ(x) = 1 for all x ∈ M with x = 1. We do this by induction on |M |, the base case |M | = 1 being clear by the hypothesis that G is residually P . Assume now that |M | > 1. If M contains only one non-trivial element, then the result is clear since G is residually P . We may therefore assume that M contains

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two non-trivial elements, say x and y. Let N be the normal closure of y in G. Thus N is the subgroup of G generated by the conjugacy class of y. Since N has a trivial centralizer by hypothesis, we have x ∈ CG (N ), so that there exists g ∈ G with [x, gyg −1 ] = 1. Let M  = M \ {x, y} ∪ {[x, gyg −1 ]}. By induction, there is a quotient map ρ : G → Q onto some group satisfying P , such that ρ(z) = 1 for all z ∈ M  with z = 1. In particular ρ(x) = 1 = ρ(y). Thus ρ(z) = 1 for all z ∈ M with z = 1, as required. If G is non-elementary hyperbolic, then EG (G) coincides with the amenable radical of G. Assume that G is acylindrically hyperbolic with trivial amenable radical. Let N be a non-trivial normal subgroup. Then N CG (N ) is acylinrically hyperbolic by [96, Corollary 1.5], so by (the proof of) [96, Lemma 7.3], either N or CG (N ) is contained in the amenable radical of G. Since the latter is trivial while N is non-trivial, we conclude that CG (N ) is trivial.  5.4

The space of marked groups

The remark preceding Proposition 5.12 can be nicely interpreted in the framework of the space of marked groups. This is a compact Hausdorff topological space that was alluded to by M. Gromov in [55], and formally defined by R. Grigorchuk [52]. In this section, we follow the presentation of Champetier–Guirardel [35, §2]. A (d-generated) marked group is a pair (G, S) consisting of a group G and a d-tuple (s1 , . . . , sd ) which generates G. Let Fd be the free group of rank d and fix a generating d-tuple (a1 , . . . , ad ). Every d-generated marked group (G, S) gives rise to a unique surjective homomorphism Fd → G with ai → si . Thus the set of isomorphism classes of d-generated marked groups can be identified with the set of normal subgroups of Fd . That set carries a natural totally disconnected compact Hausdorff topology, namely the Chabauty topology. The space of dgenerated marked groups is the compact space consisting of the isomorphism classes of d-generated marked groups. An alternative way to see it is by associating to a marked group (G, S) its Cayley graph Cay(G, S), viewed as a directed edgelabeled graph with labels in S. Any isomorphism class of d-generated marked groups admits a representative of the form (H, S), so that the Cayley graph of H with respect to S has the same degree and the same set of labels as the Cayley graph of (G, S). A basic neighbourhood of the isomorphism class of (G, S) in the space of marked groups consists of the classes admitting a respresentative (H, S) such that the n-ball around the identity in Cay(G, S) and Cay(H, S) are isomorphic as directed, edge-labeled graphs. It is customary to identify a marked group (G, S) with its isomorphism class; that abuse of language and notation should not cause any confusion. Here are a few examples of converging sequences in the space of marked groups. • The sequence (Z/nZ, {1 + nZ}) converges to (Z, {1}) as n tends to infinity in the space of cyclic marked groups. • The sequence (PSLn (Fp ), Ep ) converges to (PSLn (Z), E) as the prime p tends to infinity, where Ep and E are the images of the sets of elementary matrices.

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Caprace: Finite and infinite quotients of discrete and indiscrete groups • If G = S | r1 , r2 , . . . is an infinite presentation of a group G, then (G, S) is a limit of the sequence (Gn , S), where Gn = S | r1 , . . . , rn . The following basic fact is well known and easy to see.

Lemma 5.13 Let G be a finitely presented group. For every marking (G, S), the set {(G/N, SN/N ) | N  G} of marked quotients of G is a neighbourhood of (G, S) in the space of marked groups. Since finitely generated nilpotent groups are finitely presented, it follows that a non-trivial nilpotent group cannot be a limit of perfect groups in the space of marked groups. It is a natural problem to study the limits of non-abelian finite simple groups. We emphasize the following question. Problem 5.14 (i) Find an algebraic characterization of the limits of non-abelian finite simple groups in the space of marked groups. (ii) Can an infinite group of finite exponent be a limit of non-abelian finite simple groups? It should be emphasized that being a limit of a sequence of non-abelian finite simple groups is an algebraic property which is independent of any choice of marking. Indeed, this follows from a general observation due to Cornulier–Guyot–Pitsch [43, Lemma 1], so that Problem 5.14(i) is well-posed. A more precise version of Problem 5.14(ii) is proposed by Ivanov–Olshanskii in [64, Problem 2]. In an earlier version of these notes, I also asked whether a metabelian group can be a limit of non-abelian finite simple groups, or whether a limit of non-abelian finite simple groups can satisfy a law. The latter two questions were positively answered by Y. Cornulier, who pointed out the following. Proposition 5.15 (Y. Cornulier) For any prime p, the wreath product Cp  Z is a limit of alternating groups of prime degrees in the space of marked groups. In particular Z  Z is also a limit of alternating groups. Proof Let us start with the case where p is odd. Let q be another prime with q > 2p. Let a = (1, . . . , p) and b = (1, . . . , q) be cyclic permutations in Sym(q). Notice that G = a, b is contained in Alt(q). We claim that G = Alt(q). First observe that a and bp−1 ab−p+1 are two p-cycles whose support overlap in the singleton {p}. Therefore they generate a group preserving the set {1, . . . , 2p−1} and whose action on that set is 2-transitive by [5, Lemma 3.1]. A classical result of C. Jordan (see [125, Theorem 13.9]) ensures that a primitive group of degree k containing a prime cycle of order p ≤ k − 3 contains Alt(k). Using that fact (and a direct computation in case p = 3), we see that a, bp−1 ab−p+1 ∼ = Alt(2p − 1). It is easy to see that the group generated by the b -conjugates of Alt(2p − 1) is

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2-transitive on {1, . . . , q}. Using again Jordan’s result, it follows that this group is the full Alt(q). This proves the claim. Consider now the generating pair S = (a, t) for G, where t = bp . For n < q/p, the ball of radius n around the identity in Cay(G, S) is isomorphic to the ball of radius n in the Cayley graph of Cp  Z with respect to the natural generating pair. The desired assertion follows by letting the prime q tend to infinity. For p = 2, one defines a = (1, 2)(3, 4) and uses similar considerations. Finally, the result for Z  Z follows since the latter group is a limit of Cp  Z for p tending to infinity.  Notice that Proposition 5.15 together with Lemma 5.13 yields a proof of the well-known fact that the lamplighter group Cp  Z is not finitely presented. Problem 5.14 is quite natural in its own right. Moreover, a negative answer to Problem 5.14(ii) would yield a negative answer to Problem 5.2. In order to see that, let us recall another well known and easy fact. Lemma 5.16 Let G be a group and P be a collection of groups. Assume that G is fully residually in P. Then for any d-generator marking (G, S), there is a sequence of d-generated marked groups (Hn , Sn ) with Hn ∈ P which converges to (G, S) in the space of marked groups. In view of Corollary 5.11, we obtain: Corollary 5.17 If all hyperbolic groups are residually finite, then for every nonelementary hyperbolic group G with EG (G) = {1}, every d-generator marking (G, S) is a limit a non-abelian finite simple groups (containing arbitrarily alternating groups) in the space of d-generated marked groups. A crucial point to underline is that, in the space of marked groups, limits of hyperbolic groups can be quite wild: they can be infinite groups of finite exponent by [64]. Thus, if all hyperbolic groups were residually finite, then Burnside groups would arise as limits of non-abelian finite simple groups. 5.5

Examples of fully residually finite simple groups

Currently, rather little is known on limits of non-abelian finite simple groups in the space of marked groups. We devote this subsection to a list of a few families of groups that are fully residually finite simple. Those are thus limits of finite simple groups with respect to any choice of marking in view of Lemma 5.16. (1) The free group Fr of rank r ≥ 2 is fully residually finite simple. This can be established as follows. First observe that PSL2 (Z) has a finite index subgroup isomorphic to the free group F2 (this can be seen geometrically, or algebraically using that PSL2 (Z) is isomorphic to C2 ∗ C3 ). In particular, for every r ≥ 2, the free group Fr is a finite index subgroup of PSL2 (Z). On the other hand, for any finite set M ⊂ PSL2 (Z) and any sufficiently large prime p,

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Caprace: Finite and infinite quotients of discrete and indiscrete groups the congruence quotient map PSL2 (Z) → PSL2 (Fp ) is injective on M . Since PSL2 (Fp ) is generated by elementary matrices, which have order p, it cannot act non-trivially on a set with less than p elements. Hence PSL2 (Fp ) does not have any proper subgroup of index < p. It follows that the restriction of the quotient map PSL2 (Z) → PSL2 (Fp ) to a fixed finite index subgroup of PSL2 (Z) is surjective for all but finitely many primes p. It follows that Fr is fully residually finite simple.

(2) The free group Fr of rank r ≥ 2 is also fully residually in the class {Alt(n) | n ≥ 5} of alternating groups. Indeed, it follows from [68] that Fr is residually alternating, and the conclusion then follows from Proposition 5.12. See also [46] and [129] for various extensions of that result. (3) In view of (1), any group that is fully residually free must also be fully residually finite simple. Fully residually free groups coincide with the so-called limit groups in the sense of Z. Sela, see [115]. That class includes the fundamental groups of closed surfaces of genus ≥ 2. (4) Let G = ∗i∈I Gi be a free product of at least two non-trivial residually finite groups. It is proved in [118] that G is residually in {Alt(n) | n ≥ 5} unless G is the infinite dihedral group. Hence G is fully residually in {Alt(n) | n ≥ 5} by Proposition 5.12. Various extensions of that result were established in [79, 78]. (5) The fundamental group G of a closed 3-manifold is fully residually finite simple if it is infinite and contained in SL2 (Q). This follows from [80] and Proposition 5.12. (6) For n ≥ 3, the group Out(Fn ) is residually in {Alt(n) | n ≥ 5} by [50], hence fully residually alternating by Proposition 5.12 and [96, §8(b)]. 5.6

Virtual specialties, after I. Agol, F. Haglund and D. Wise

The concept of virtually special groups was introduced by Haglund–Wise [58]. We do not reproduce the definition here; the reader may consult the excellent surveys [12], [4], [111] and [128], that cover the basic theory of CAT(0) cube complexes and virtually special groups, and its relevance to the theory of 3-manifolds. The following result is one of the most spectacular breakthroughs in Geometric Group Theory over the past decade. Theorem 5.18 Let G be a non-elementary hyperbolic group. (i) (I. Agol [1]) If G is capable of acting properly and cocompactly on a CAT(0) cube complex, then G is virtually special. (ii) (Haglund–Wise [58]) If G is virtually special, then: • G is linear over Z, hence residually finite; • G has a finite index subgroup that maps onto a non-abelian free group;

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• Every quasi-convex subgroup of G is separable. Even without bearing in mind what the formal definition of virtually special groups is, the key feature of Theorem 5.18 should be transparent: it provides a purely geometric condition on a group G ensuring that G enjoys very strong algebraic properties. It is also important to underline that many hyperbolic groups satisfy that geometric condition: this is notably the case of all examples mentioned in Section 5.1, except for the fundamental groups of certain negatively curved closed manifolds. In fact, at the time of this writing, the only known obstruction for an infinite hyperbolic group to act properly and cocompactly on a CAT(0) cube complex is provided by Kazhdan’s property (T) (see [92]). The strikingly broad scope of Theorem 5.18 sheds new light on Problem 5.2. Since the publication of the original paper of Haglund–Wise [58] where virtually special groups were introduced, numerous constructions and results valid in the category of hyperbolic groups were also established in the virtually special framework: Rips’ construction [58, Theorem 10.1], the Combination Theorem [59, Theorem 1.2], the Dehn Filling Theorem [133, Theorem A], [3]. The following question is thus very natural.

Problem 5.19 (i) Is every non-elementary virtually special hyperbolic group G with EG (G) = {1} fully residually finite simple? Fully residually alternating? (ii) Is it true that any two non-elementary virtually special hyperbolic groups have a common non-elementary virtually special quotient? (iii) More generally, is Olshanskii’s Theorem 5.7 valid in the virtually special ˜ with the extra property framework, i.e., can one find a quotient group G of being virtually special if the initial group G is assumed virtually special? That Problem 5.19(i) holds in the special case of free amalgamated products of finite groups is closely related to a conjecture of Dˇzambi´c–Jones [47, p. 206]. A positive solution to these questions would imply the validity of Corollaries 5.8, 5.9, 5.10, 5.11 and 5.17 in the context of virtually special groups. In particular, if Problem 5.19(iii) had a positive solution, then (i) and (ii) would also do. Such a result would be highly interesting from the point of view of hyperbolic groups, as well as from that of asymptotic properties of the finite simple groups. Coming back to Problem 5.2, the previous discussion suggests that the most promising candidates of non-residually finite hyperbolic groups are to be found among the infinite hyperbolic groups with Kazhdan’s property (T), since those are not virtually special. Here is an explicit presentation of such a group, where [x, y] = xyx−1 y −1 .

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Example 5.20 The group E = x, y, z, t, r | x7 , y 7 , [x, y]z −1 , [x, z], [y, z], t2 , r73 , trtr, [x2 yz −1 , t], [xyz 3 , tr], [x3 yz 2 , tr17 ], [x, tr−34 ], [y, tr−32 ], [z, tr−29 ], [x−2 yz, tr−25 ], [x−1 yz −3 , tr−19 ], [x−3 yz −2 , tr−11 ] , is an infinite Gromov hyperbolic group enjoying Kazhdan’s property (T), see [26, Theorem 1]. The group E has a retraction E → x, y that is trivial on t, r , and a retraction E → t, r that is trivial on x, y . Their product is a quotient homomorphism of E onto the direct product of the Heisenberg group over F7 with the dihedral group of order 146; the kernel of that map is torsion-free (see [26, Theorem 1]). I do not know any larger finite quotient of E. In particular, I do not know any non-abelian finite simple quotient of E. In view of Corollary 5.9, a negative solution to the following problem would imply the existence of a non-residually finite hyperbolic group. Problem 5.21 Does the group E admit finite simple quotients containing arbitrarily large alternating groups? Is E residually finite simple?

Acknowledgements I thank the Isaac Newton Institute for Mathematical Sciences, Cambridge for support and hospitality during the programme Non-positive curvature group actions and cohomology where part of the work on this paper was accomplished. The paper is based on mini-courses I gave at the university of Lille in March, at the Winter School on Arithmetic Groups in Ein Gedi in March, at the Newton Institute in April, and at the Groups St Andrews conference in Birmingham in August 2017. I thank the organizers of those events for their hospitality, and the participants for their feedback from which I benefited greatly. I am especially grateful to V. Guirardel, P. Neumann, C. Praeger, N. Radu for inspiring conversations, and to Y. Cornulier, J.-P. Tignol and an anonymous referee for their comments and corrections on an earlier version of these notes. Finally, it is a pleasure to acknowledge an intellectual debt to Marc Burger, Shahar Mozes and Dani Wise: my interest in the topic of these notes grew out of my fascination for their seminal contributions. References [1] Ian Agol, The virtual Haken conjecture, Doc. Math. 18 (2013), 1045–1087, With an appendix by Agol, Daniel Groves, and Jason Manning. MR 3104553 [2] Ian Agol, Daniel Groves, and Jason Fox Manning, Residual finiteness, QCERF and fillings of hyperbolic groups, Geom. Topol. 13 (2009), no. 2, 1043–1073. MR 2470970 [3] Ian Agol, Daniel Groves, and Jason Fox Manning, An alternate proof of Wise’s malnormal special quotient theorem, Forum Math. Pi 4 (2016), e1, 54 pp. MR 3456181

Caprace: Finite and infinite quotients of discrete and indiscrete groups

63

[4] Matthias Aschenbrenner, Stefan Friedl, and Henry Wilton, 3-manifold groups, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Z¨ urich, 2015. MR 3444187 [5] Uri Bader, Pierre-Emmanuel Caprace, Tsachik Gelander, and Shahar Mozes, Simple groups without lattices, Bull. Lond. Math. Soc. 44 (2012), no. 1, 55–67. MR 2881324 [6] Uri Bader, Pierre-Emmanuel Caprace, and Jean L´ecureux, On the linearity of lattices in affine buildings and ergodicity of the singular cartan flow, Preprint, arXiv: 1608.06265, 2016. [7] Uri Bader and Yehuda Shalom, Factor and normal subgroup theorems for lattices in products of groups, Invent. Math. 163 (2006), no. 2, 415–454. MR 2207022 [8] Hyman Bass and Alexander Lubotzky, Tree lattices, Progress in Mathematics, vol. 176, Birkh¨auser Boston, Inc., Boston, MA, 2001, With appendices by Bass, L. Carbone, Lubotzky, G. Rosenberg and J. Tits. MR 1794898 [9] Gilbert Baumslag, A non-cyclic one-relator group all of whose finite quotients are cyclic, J. Austral. Math. Soc. 10 (1969), 497–498. MR 0254127 [10] Bachir Bekka, Pierre de la Harpe, and Alain Valette, Kazhdan’s property (T), New Mathematical Monographs, vol. 11, Cambridge University Press, Cambridge, 2008. MR 2415834 [11] Mikhail Belolipetsky, Counting maximal arithmetic subgroups, Duke Math. J. 140 (2007), no. 1, 1–33, With an appendix by Jordan Ellenberg and Akshay Venkatesh. MR 2355066 [12] Mladen Bestvina, Geometric group theory and 3-manifolds hand in hand: the fulfillment of Thurston’s vision, Bull. Amer. Math. Soc. (N.S.) 51 (2014), no. 1, 53–70. MR 3119822 [13] Meenaxi Bhattacharjee, Constructing finitely presented infinite nearly simple groups, Comm. Algebra 22 (1994), no. 11, 4561–4589. MR 1284345 [14] Collin Bleak and Martyn Quick, The infinite simple group V of Richard J. Thompson: presentations by permutations, Groups Geom. Dyn. 11 (2017), no. 4, 1401–1436. [15] Ievgen Bondarenko, Daniele D’Angeli, and Emanuele Rodaro, The lamplighter group Z3  Z generated by a bireversible automaton, Comm. Algebra 44 (2016), no. 12, 5257–5268. MR 3520274 [16] Ievgen Bondarenko and Bohdan Kivva, Automaton groups and complete square complexes, Preprint arXiv:1707.00215. [17] Martin R. Bridson and Andr´e Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486 [18] Kenneth S. Brown, The geometry of finitely presented infinite simple groups, Algorithms and classification in combinatorial group theory (Berkeley, CA, 1989), Math. Sci. Res. Inst. Publ., vol. 23, Springer, New York, 1992, pp. 121–136. MR 1230631 [19] M. Burger, T. Gelander, A. Lubotzky, and S. Mozes, Counting hyperbolic manifolds, Geom. Funct. Anal. 12 (2002), no. 6, 1161–1173. MR 1952926 [20] Marc Burger and Shahar Mozes, Finitely presented simple groups and products of trees, C. R. Acad. Sci. Paris S´er. I Math. 324 (1997), no. 7, 747–752. MR 1446574 [21] Marc Burger and Shahar Mozes, Groups acting on trees: from local to global structure, ´ Inst. Hautes Etudes Sci. Publ. Math. (2000), no. 92, 113–150 (2001). MR 1839488 ´ [22] Marc Burger and Shahar Mozes, Lattices in product of trees, Inst. Hautes Etudes Sci. Publ. Math. (2000), no. 92, 151–194 (2001). MR 1839489 [23] Marc Burger, Shahar Mozes, and Robert J. Zimmer, Linear representations and arithmeticity of lattices in products of trees, Essays in geometric group theory, Ramanujan Math. Soc. Lect. Notes Ser., vol. 9, Ramanujan Math. Soc., Mysore, 2009, pp. 1–25. MR 2605353

64

Caprace: Finite and infinite quotients of discrete and indiscrete groups

[24] Ruth Camm, Simple free products, J. London Math. Soc. 28 (1953), 66–76. MR 0052420 [25] J. W. Cannon, W. J. Floyd, and W. R. Parry, Introductory notes on Richard Thompson’s groups, Enseign. Math. (2) 42 (1996), no. 3-4, 215–256. MR 1426438 [26] Pierre-Emmanuel Caprace, A sixteen-relator presentation of an infinite hyperbolic Kazhdan group, Preprint arXiv:1708.09772, 2017. [27] Pierre-Emmanuel Caprace, Yves Cornulier, Nicolas Monod, and Romain Tessera, Amenable hyperbolic groups, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 11, 2903– 2947. MR 3420526 [28] Pierre-Emmanuel Caprace and Koji Fujiwara, Rank-one isometries of buildings and quasi-morphisms of Kac-Moody groups, Geom. Funct. Anal. 19 (2010), no. 5, 1296– 1319. MR 2585575 [29] Pierre-Emmanuel Caprace, Peter H. Kropholler, Colin D. Reid, and Phillip Wesolek, On the residual and profinite closures of commensurated subgroups, Preprint arXiv: 1706.06853. [30] Pierre-Emmanuel Caprace and Adrien Le Boudec, Bounding the covolume of lattices in products, Preprint arXiv:1805.04469. [31] Pierre-Emmanuel Caprace and Nicolas Monod, Decomposing locally compact groups into simple pieces, Math. Proc. Cambridge Philos. Soc. 150 (2011), no. 1, 97–128. MR 2739075 [32] Pierre-Emmanuel Caprace and Nicolas Monod, A lattice in more than two Kac-Moody groups is arithmetic, Israel J. Math. 190 (2012), 413–444. MR 2956249 [33] Pierre-Emmanuel Caprace and Bertrand R´emy, Simplicity and superrigidity of twin building lattices, Invent. Math. 176 (2009), no. 1, 169–221. MR 2485882 [34] Pierre-Emmanuel Caprace and Phillip Wesolek, Indicability, residual finiteness, and simple subquotients of groups acting on trees, to appear in Geom. Topol. [35] Christophe Champetier and Vincent Guirardel, Limit groups as limits of free groups, Israel J. Math. 146 (2005), 1–75. MR 2151593 [36] Ted Chinburg, Holley Friedlander, Sean Howe, Michiel Kosters, Bhairav Singh, Matthew Stover, Ying Zhang, and Paul Ziegler, Presentations for quaternionic S-unit groups, Exp. Math. 24 (2015), no. 2, 175–182. MR 3350524 [37] Ted Chinburg and Matthew Stover, Small generators for S-unit groups of division algebras, New York J. Math. 20 (2014), 1175–1202. MR 3291615 [38] M. Coornaert, T. Delzant, and A. Papadopoulos, G´eom´etrie et th´eorie des groupes, Lecture Notes in Mathematics, vol. 1441, Springer-Verlag, Berlin, 1990, Les groupes hyperboliques de Gromov. [Gromov hyperbolic groups], With an English summary. MR 1075994 [39] Kevin Corlette, Archimedean superrigidity and hyperbolic geometry, Ann. of Math. (2) 135 (1992), no. 1, 165–182. MR 1147961 [40] F. Dahmani, V. Guirardel, and D. Osin, Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces, Mem. Amer. Math. Soc. 245 (2017), no. 1156, v+152. MR 3589159 [41] Giuliana Davidoff, Peter Sarnak, and Alain Valette, Elementary number theory, group theory, and Ramanujan graphs, London Mathematical Society Student Texts, vol. 55, Cambridge University Press, Cambridge, 2003. MR 1989434 [42] Michael W. Davis, The geometry and topology of Coxeter groups, Introduction to modern mathematics, Adv. Lect. Math. (ALM), vol. 33, Int. Press, Somerville, MA, 2015, pp. 129–142. MR 3445448 [43] Yves de Cornulier, Luc Guyot, and Wolfgang Pitsch, On the isolated points in the space of groups, J. Algebra 307 (2007), no. 1, 254–277. MR 2278053 [44] Thomas Delzant, Sous-groupes distingu´es et quotients des groupes hyperboliques,

Caprace: Finite and infinite quotients of discrete and indiscrete groups

65

Duke Math. J. 83 (1996), no. 3, 661–682. MR 1390660 [45] Martin Deraux, John R. Parker, and Julien Paupert, New non-arithmetic complex hyperbolic lattices, Invent. Math. 203 (2016), no. 3, 681–771. MR 3461365 ´ [46] John D. Dixon, L´ aszl´ o Pyber, Akos Seress, and Aner Shalev, Residual properties of free groups and probabilistic methods, J. Reine Angew. Math. 556 (2003), 159–172. MR 1971144 [47] Amir Dˇzambi´c and Gareth A. Jones, p-adic Hurwitz groups, J. Algebra 379 (2013), 179–207. MR 3019251 [48] Tsachik Gelander and Arie Levit, Counting commensurability classes of hyperbolic manifolds, Geom. Funct. Anal. 24 (2014), no. 5, 1431–1447. MR 3261631 ´ Ghys and P. de la Harpe (eds.), Sur les groupes hyperboliques d’apr`es Mikhael [49] E. Gromov, Progress in Mathematics, vol. 83, Birkh¨ auser Boston, Inc., Boston, MA, 1990, Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988. MR 1086648 [50] Robert Gilman, Finite quotients of the automorphism group of a free group, Canad. J. Math. 29 (1977), no. 3, 541–551. MR 0435226 [51] Yair Glasner and Shahar Mozes, Automata and square complexes, Geom. Dedicata 111 (2005), 43–64. MR 2155175 [52] R. I. Grigorchuk, Degrees of growth of finitely generated groups and the theory of invariant means, Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), no. 5, 939–985. MR 764305 [53] R. I. Grigorchuk, Just infinite branch groups, New horizons in pro-p groups, Progr. Math., vol. 184, Birkh¨ auser Boston, Boston, MA, 2000, pp. 121–179. MR 1765119 [54] M. Gromov, Hyperbolic manifolds, groups and actions, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 183–213. MR 624814 [55] Mikhael Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes ´ Etudes Sci. Publ. Math. (1981), no. 53, 53–73. MR 623534 [56] M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829 [57] Mikhail Gromov and Richard Schoen, Harmonic maps into singular spaces and p´ adic superrigidity for lattices in groups of rank one, Inst. Hautes Etudes Sci. Publ. Math. (1992), no. 76, 165–246. MR 1215595 [58] Fr´ed´eric Haglund and Daniel T. Wise, Special cube complexes, Geom. Funct. Anal. 17 (2008), no. 5, 1551–1620. MR 2377497 [59] Fr´ed´eric Haglund and Daniel T. Wise, A combination theorem for special cube complexes, Ann. of Math. (2) 176 (2012), no. 3, 1427–1482. MR 2979855 [60] Graham Higman, A finitely generated infinite simple group, J. London Math. Soc. 26 (1951), 61–64. MR 0038348 [61] Graham Higman, Finitely presented infinite simple groups, Department of Pure Mathematics, Department of Mathematics, I.A.S. Australian National University, Canberra, 1974, Notes on Pure Mathematics, No. 8 (1974). MR 0376874 [62] Graham Higman, B. H. Neumann, and Hanna Neumann, Embedding theorems for groups, J. London Math. Soc. 24 (1949), 247–254. MR 0032641 [63] Tim Hsu and Daniel T. Wise, Embedding theorems for non-positively curved polygons of finite groups, J. Pure Appl. Algebra 123 (1998), no. 1-3, 201–221. MR 1492901 [64] S. V. Ivanov and A. Yu. Ol’shanski˘ı, Hyperbolic groups and their quotients of bounded exponents, Trans. Amer. Math. Soc. 348 (1996), no. 6, 2091–2138. MR 1327257 [65] Nathan Jacobson, Basic algebra. II, second ed., W. H. Freeman and Company, New York, 1989. MR 1009787

66

Caprace: Finite and infinite quotients of discrete and indiscrete groups

[66] David Janzen and Daniel T. Wise, A smallest irreducible lattice in the product of trees, Algebr. Geom. Topol. 9 (2009), no. 4, 2191–2201. MR 2558308 [67] Ilya Kapovich and Daniel T. Wise, The equivalence of some residual properties of word-hyperbolic groups, J. Algebra 223 (2000), no. 2, 562–583. MR 1735163 [68] Robert A. Katz and Wilhelm Magnus, Residual properties of free groups, Comm. Pure Appl. Math. 22 (1968), 1–13. MR 0233873 [69] Jason S. Kimberley and Guyan Robertson, Groups acting on products of trees, tiling systems and analytic K-theory, New York J. Math. 8 (2002), 111–131. MR 1923572 [70] Carlisle S. H. King, Generation of finite simple groups by an involution and an element of prime order, J. Algebra 478 (2017), 153–173. MR 3621666 [71] Martin Kneser, Orthogonale Gruppen u ¨ber algebraischen Zahlk¨ orpern, J. Reine Angew. Math. 196 (1956), 213–220. MR 0080101 [72] Peter K¨ohler, Thomas Meixner, and Michael Wester, The affine building of type A˜2 over a local field of characteristic two, Arch. Math. (Basel) 42 (1984), no. 5, 400–407. MR 756691 [73] Peter K¨ohler, Triangle groups, Comm. Algebra 12 (1984), no. 13-14, 1595–1625. MR 743306 [74] Peter K¨ohler, The 2-adic affine building of type A˜2 and its finite projections, J. Combin. Theory Ser. A 38 (1985), no. 2, 203–209. MR 784716 [75] A. G. Kuroˇs, Teoriya Grupp, OGIZ, Moscow-Leningrad, 1944. MR 0022843 [76] Adrien Le Boudec, Groups acting on trees with almost prescribed local action, Comment. Math. Helv. 91 (2016), no. 2, 253–293. MR 3493371 [77] Martin W. Liebeck, Probabilistic and asymptotic aspects of finite simple groups, Probabilistic group theory, combinatorics, and computing, Lecture Notes in Math., vol. 2070, Springer, London, 2013, pp. 1–34. MR 3026185 [78] Martin W. Liebeck and Aner Shalev, Residual properties of free products of finite groups, J. Algebra 268 (2003), no. 1, 286–289. MR 2005288 [79] Martin W. Liebeck and Aner Shalev, Residual properties of the modular group and other free products, J. Algebra 268 (2003), no. 1, 264–285. MR 2005287 [80] D. D. Long and A. W. Reid, Simple quotients of hyperbolic 3-manifold groups, Proc. Amer. Math. Soc. 126 (1998), no. 3, 877–880. MR 1459136 [81] Darren Long and Alan W. Reid, Surface subgroups and subgroup separability in 3manifold topology, Publica¸c˜ oes Matem´ aticas do IMPA. [IMPA Mathematical Publications], Instituto Nacional de Matem´ atica Pura e Aplicada (IMPA), Rio de Janeiro, 2005, 25o Col´oquio Brasileiro de Matem´atica. [25th Brazilian Mathematics Colloquium]. MR 2164951 [82] Alexander Lubotzky, Some more non-arithmetic rigid groups, Geometry, spectral theory, groups, and dynamics, Contemp. Math., vol. 387, Amer. Math. Soc., Providence, RI, 2005, pp. 237–244. MR 2180210 [83] G. A. Margulis, Finiteness of quotient groups of discrete subgroups, Funktsional. Anal. i Prilozhen. 13 (1979), no. 3, 28–39. MR 545365 [84] G. A. Margulis, Multiplicative groups of a quaternion algebra over a global field, Dokl. Akad. Nauk SSSR 252 (1980), no. 3, 542–546. MR 577836 [85] G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17, Springer-Verlag, Berlin, 1991. MR 1090825 [86] Alexandre Martin, On the cubical geometry of Higman’s group, Duke Math. J. 166 (2017), no. 4, 707–738. MR 3619304 [87] Ashot Minasyan and Denis Osin, Acylindrical hyperbolicity of groups acting on trees, Math. Ann. 362 (2015), no. 3-4, 1055–1105. MR 3368093 [88] Shahar Mozes, A zero entropy, mixing of all orders tiling system, Symbolic dynamics

Caprace: Finite and infinite quotients of discrete and indiscrete groups

[89] [90] [91]

[92] [93] [94] [95] [96] [97] [98]

[99] [100] [101] [102] [103] [104]

[105]

[106]

[107] [108] [109] [110]

67

and its applications (New Haven, CT, 1991), Contemp. Math., vol. 135, Amer. Math. Soc., Providence, RI, 1992, pp. 319–325. MR 1185097 B. H. Neumann, Some remarks on infinite groups., J. Lond. Math. Soc. 12 (1937), 120–127. B. H. Neumann and Hanna Neumann, A contribution to the embedding theory of group amalgams, Proc. London Math. Soc. (3) 3 (1953), 243–256. MR 0057880 Peter M. Neumann, The SQ-universality of some finitely presented groups, J. Austral. Math. Soc. 16 (1973), 1–6, Collection of articles dedicated to the memory of Hanna Neumann, I. MR 0333017 Graham Niblo and Lawrence Reeves, Groups acting on CAT(0) cube complexes, Geom. Topol. 1 (1997), 1–7. MR 1432323 A. Yu. Ol’shanski˘ı, On residualing homomorphisms and G-subgroups of hyperbolic groups, Internat. J. Algebra Comput. 3 (1993), no. 4, 365–409. MR 1250244 A. Yu. Ol’shanski˘ı, SQ-universality of hyperbolic groups, Mat. Sb. 186 (1995), no. 8, 119–132. MR 1357360 A. Yu. Ol’shanski˘ı, On the Bass-Lubotzky question about quotients of hyperbolic groups, J. Algebra 226 (2000), no. 2, 807–817. MR 1752761 D. Osin, Acylindrically hyperbolic groups, Trans. Amer. Math. Soc. 368 (2016), no. 2, 851–888. MR 3430352 V. P. Platonov, Arithmetic and structural problems in linear algebraic groups, pp. 471–478, Anad. Math. Ongress, Montreal, Que., 1975. MR 0466334 Vladimir Platonov and Andrei Rapinchuk, Algebraic groups and number theory, Pure and Applied Mathematics, vol. 139, Academic Press, Inc., Boston, MA, 1994, Translated from the 1991 Russian original by Rachel Rowen. MR 1278263 Nicolas Radu, New simple lattices in products of trees and their projections, Preprint arXiv:1712.01091. Nicolas Radu, A classification theorem for boundary 2-transitive automorphism groups of trees, Invent. Math. 209 (2017), no. 1, 1–60. MR 3660305 Nicolas Radu, PhD thesis, Universit´e catholique de Louvain, In preparation, 2017. M. S. Raghunathan, Discrete subgroups of Lie groups, Math. Student (2007), Special Centenary Volume, 59–70 (2008). MR 2527560 Andrei S. Rapinchuk, The Margulis-Platonov conjecture for SL1,D and 2-generation of finite simple groups, Math. Z. 252 (2006), no. 2, 295–313. MR 2207799 Andrei Rapinchuk and Alexander Potapchik, Normal subgroups of SL1,D and the classification of finite simple groups, Proc. Indian Acad. Sci. Math. Sci. 106 (1996), no. 4, 329–368. MR 1425612 Andrei S. Rapinchuk, Yoav Segev, and Gary M. Seitz, Finite quotients of the multiplicative group of a finite dimensional division algebra are solvable, J. Amer. Math. Soc. 15 (2002), no. 4, 929–978. MR 1915823 Diego Rattaggi, Computations in groups acting on a product of trees: Normal subgroup structures and quaternion lattices, ProQuest LLC, Ann Arbor, MI, 2004, Thesis (Dr.sc.math.)–Eidgenoessische Technische Hochschule Zuerich (Switzerland). MR 2715704 Diego Rattaggi, Anti-tori in square complex groups, Geom. Dedicata 114 (2005), 189–207. MR 2174099 Diego Rattaggi, A finitely presented torsion-free simple group, J. Group Theory 10 (2007), no. 3, 363–371. MR 2320973 M. A. Ronan, Triangle geometries, J. Combin. Theory Ser. A 37 (1984), no. 3, 294– 319. MR 769219 Nithi Rungtanapirom, Quaternionic arithmetic lattices of rank 2 and a fake quadric in characteristic 2, Preprint arXiv:1707.09925.

68

Caprace: Finite and infinite quotients of discrete and indiscrete groups

[111] Michah Sageev, CAT(0) cube complexes and groups, Geometric group theory, IAS/Park City Math. Ser., vol. 21, Amer. Math. Soc., Providence, RI, 2014, pp. 7–54. MR 3329724 [112] Paul E. Schupp, Small cancellation theory over free products with amalgamation, Math. Ann. 193 (1971), 255–264. MR 0291298 [113] Paul E. Schupp, A survey of small cancellation theory, 569–589. Studies in Logic and the Foundations of Math., Vol. 71. MR 0412289 [114] Yoav Segev and Gary M. Seitz, Anisotropic groups of type An and the commuting graph of finite simple groups, Pacific J. Math. 202 (2002), no. 1, 125–225. MR 1883974 [115] Zlil Sela, Diophantine geometry over groups. I. Makanin-Razborov diagrams, Publ. ´ Math. Inst. Hautes Etudes Sci. (2001), no. 93, 31–105. MR 1863735 [116] Yehuda Shalom, Rigidity of commensurators and irreducible lattices, Invent. Math. 141 (2000), no. 1, 1–54. MR 1767270 [117] Jakob Stix and Alina Vdovina, Simply transitive quaternionic lattices of rank 2 over Fq (t) and a non-classical fake quadric, Math. Proc. Cambridge Philos. Soc. 163 (2017), no. 3, 453–498. MR 3708519 [118] Chiara Tamburini and John S. Wilson, A residual property of certain free products, Math. Z. 186 (1984), no. 4, 525–530. MR 744963 [119] Jacques Tits, Buildings and group amalgamations, Proceedings of groups—St. Andrews 1985, London Math. Soc. Lecture Note Ser., vol. 121, Cambridge Univ. Press, Cambridge, 1986, pp. 110–127. MR 896503 [120] Jacques Tits, R´esum´es des cours au Coll`ege de France 1973–2000, Documents Math´ematiques (Paris) [Mathematical Documents (Paris)], vol. 12, Soci´et´e Math´ematique de France, Paris, 2013. MR 3235648 [121] V. I. Trofimov and R. M. Weiss, Graphs with a locally linear group of automorphisms, Math. Proc. Cambridge Philos. Soc. 118 (1995), no. 2, 191–206. MR 1341785 [122] Marie-France Vign´eras, Arithm´etique des alg`ebres de quaternions, Lecture Notes in Mathematics, vol. 800, Springer, Berlin, 1980. MR 580949 [123] Hsien Chung Wang, Topics on totally discontinuous groups, 459–487. Pure and Appl. Math., Vol. 8. MR 0414787 [124] Richard Weiss, Groups with a (B, N )-pair and locally transitive graphs, Nagoya Math. J. 74 (1979), 1–21. MR 535958 [125] Helmut Wielandt, Finite permutation groups, Translated from the German by R. Bercov, Academic Press, New York-London, 1964. MR 0183775 [126] John S. Wilson, Groups with every proper quotient finite, Proc. Cambridge Philos. Soc. 69 (1971), 373–391. MR 0274575 [127] John S. Wilson, On just infinite abstract and profinite groups, New horizons in pro-p groups, Progr. Math., vol. 184, Birkh¨auser Boston, Boston, MA, 2000, pp. 181–203. MR 1765120 [128] Henry Wilton, Non-positively curved cube complexes, Course notes, available as AMS Open Math Notes:201704.110697, 2011. [129] Henry Wilton, Alternating quotients of free groups, Enseign. Math. (2) 58 (2012), no. 1-2, 49–60. MR 2985009 [130] Daniel T. Wise, Non-positively curved squared complexes: Aperiodic tilings and nonresidually finite groups, ProQuest LLC, Ann Arbor, MI, 1996, Thesis (Ph.D.)– Princeton University. MR 2694733 [131] Daniel T. Wise, Subgroup separability of the figure 8 knot group, Topology 45 (2006), no. 3, 421–463. MR 2218750 [132] Daniel T. Wise, Complete square complexes, Comment. Math. Helv. 82 (2007), no. 4, 683–724. MR 2341837

Caprace: Finite and infinite quotients of discrete and indiscrete groups

69

[133] Daniel T. Wise, The structure of groups with a quasi-convex hierarchy, Preprint available at https://www.math.u-psud.fr/~haglund/Hierarchy29Feb2012.pdf, 2012.

LOCAL-GLOBAL CONJECTURES AND BLOCKS OF SIMPLE GROUPS RADHA KESSAR∗ and GUNTER MALLE† ∗

Department of Mathematics, City, University of London, Northampton Square, EC1V 0HB London, U.K. Email: [email protected]

† FB Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany. Email: [email protected]

Abstract We give an expanded treatment of our lecture series at the 2017 Groups St Andrews conference in Birmingham on local-global conjectures and the block theory of finite reductive groups.

1

Introduction

The aim of these notes is to describe recent progress in the ordinary and modular representation theory of finite groups, in particular pertaining to the local-global counting conjectures. As this relies heavily on having sufficient knowledge about the representation theory of finite simple groups, we will also highlight the major advances and results obtained in the block theory of finite groups of Lie type. The general setup will be as follows: G will be a finite group, Irr(G) = {trace functions of irreducible representations G → GLn (C)} its set of irreducible complex characters. For χ ∈ Irr(G), the value χ(1) at the identity element of G is called its degree; it is the degree of any representation affording this character. We also choose a prime p (with the interesting case being the one when p divides the group order |G|). The aim is now to link, as much as possible, aspects of the representation theory of G, like its set of irreducible characters Irr(G), their degrees, and so on, to those of local subgroups of G. Here a subgroup N of G is called p-local if N = NG (Q) for some p-subgroup 1 = Q ≤ G. An important example of local subgroups is given by the normalisers NG (P ) of Sylow p-subgroups P ∈ Sylp (G).

2

The fundamental conjectures

The character theory of finite groups was invented by G. Frobenius more than a hundred years ago. But still there are many basic open questions that remain This article was written while the authors were visiting the Mathematical Sciences Research Institute in Berkeley, California in Spring 2018 for the programme Group Representation Theory and Applications supported by the National Science Foundation under Grant No. DMS-1440140.

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71

unsolved to the present day. We present some of these in this section. 2.1

The McKay conjecture

John McKay in the beginning of the 1970s counted irreducible characters of odd degree of the newly discovered sporadic simple groups; here are some such numbers: M11 : 4, M12 : 8, Co1 , F i2 : 32, B, M : 64. Strikingly, all of these are 2-powers. In 1971 Ian Macdonald [53] showed that the number of odd degree irreducible characters is a power of 2 for all symmetric groups Sn . (But obviously this statement cannot be true for all groups, think of the cyclic group of order 3.) The right generalisation seems to be as follows: let Irrp (G) := {χ ∈ Irr(G) | χ(1) ≡ 0

(mod p)}

be the subset of irreducible characters of G of degree prime to p, then the following should be true [61]: Conjecture 2.1 (McKay (1972)) Let G be a finite group, p a prime and P ∈ Sylp (G). Then | Irrp (G)| = | Irrp (NG (P ))|. This conjecture does indeed predict global data in terms of local information. Since it proposes an explicit formula for | Irrp (G)| it is sometimes also called a counting conjecture. Note that for G a sporadic group as above, or a symmetric group, Sylow 2-subgroups are self-normalising and then Irr2 (NG (P )) = Irr2 (P ) = Irr(P/P  ) is an abelian 2-group and hence of order a power of 2. Marty Isaacs [42] showed in 1973 that Conjecture 2.1 is true for all groups of odd order, using the Glauberman correspondence, before even having been aware of McKay’s paper. Example 2.2 Let G = Sn the symmetric group of degree n. Frobenius showed how the irreducible characters of G can be naturally labelled by partitions λ  n of n. Let us write χλ for the irreducible character labelled by λ. The degree of χλ is given by the well-known hook formula χλ (1) = 

n! h (h)

,

where the product runs over all hooks of λ, that is, all boxes (i, j) of the Young diagram of λ, and (h) denotes the length of the hook in that diagram starting at box (i, j). Macdonald [53] determined when this expression is an odd number, and obtained the following result: write n = 2k1 + 2k2 + . . . with k1 < k2 < . . .. Then | Irr2 (Sn )| = 2k1 +k2 +... , which is indeed a power of 2. Formulas for general p ≥ 2 can be given in terms of the p-adic expansion of n using generating functions (see [53]).

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Kessar, Malle: Local-global conjectures and blocks

On the other hand a Sylow 2-subgroup P of Sn is a direct product P = P1 × P2 × · · · with Pi = C2  C2  · · · (ki factors), an iterated wreath product, and it is self-normalising. Now as already pointed out above, the only p -characters of p-groups are the linear characters, so that Irrp (P ) = Irr(P/P  ). But | Irr(P/P  )| = | Irr(P1 /P1 )| × | Irr(P2 /P2 )| × · · · , and | Irr(Pi /Pi )| = |Pi /Pi | = 2ki , so indeed we find | Irr2 (P )| = 2k1 +k2 +... = | Irr2 (G)| as predicted by McKay’s Conjecture 2.1. While McKay’s conjecture is still open, various refinements and extensions have been proposed; as one example let us mention [45]: Conjecture 2.3 (Isaacs–Navarro (2002)) In the situation of Conjecture 2.1 there exists a bijection Ω : Irrp (G) → Irrp (NG (P )) such that Ω(χ)(1) ≡ ±χ(1) (mod p). Paul Fong [34] showed that this refinement holds for G = Sn and all primes (note that it is stronger than the original McKay conjecture only when p ≥ 5). Alexandre Turull [83] showed that it holds for all solvable groups. This was shown to hold for alternating groups by Nath [65] Another such refinement, which is currently being studied quite intensely, was proposed by Navarro [66]; it proposes that the bijection Ω should also be equivariant with respect to Gal(Qp /Qp ); see Brunat and Nath [16] for the case of alternating groups, and Ruhstorfer [74] for groups of Lie type in their defining characteristic. 2.2

The local-global conjectures

The McKay conjecture is concerned with the characters of p -degree. Now what about characters of degree divisible by p? How to relate these to local data? There is a natural extension of McKay’s conjecture, but in order to formulate this, we need to introduce p-blocks. Let O ≥ Zp be a big enough extension, for example containing all |G|th roots of unity, and decompose the group ring of G over O into a direct sum of minimal 2-sided ideals OG = B1 ⊕ . . . ⊕ Br , called the p-blocks of G. It is easily seen that this induces a partition Irr(G) = Irr(B1 )  . . .  Irr(Br ), by decreeing that χ ∈ Irr(G) lies in Irr(Bi ) if and only if χ|Bi = 0. This block subdivision can in fact be read off from the character table of G: χ, ψ ∈ Irr(G) lie in the same p-block if and only if |xG |χ(x) |xG |ψ(x) ≡ χ(1) ψ(1)

(mod P)

where P  O is the maximal ideal containing p.

for all x ∈ G,

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Richard Brauer showed how to associate to any p-block B of G a p-subgroup D ≤ G of G, unique up to conjugation, called defect group of B. This can be defined as follows: D is minimal amongst p-subgroups P of G for which there exists a p -element x of G such that P is a Sylow p-subgroup of CG (x) and |xG |χ(x) ≡ 0 χ(1)

(mod P)

for all χ ∈ Irr(B).

Brauer also constructed a block b of NG (D) called Brauer correspondent of B. The Brauer correspondent of B in NG (D) is the unique p-block b of NG (D) with defect group D such that |xG |χ(x) |xG |θG (x) ≡ χ(1) θ(1)

(mod P)

for all x ∈ G, χ ∈ Irr(B) and θ ∈ Irr(b).

Example 2.4 (a) Let G be a p-group. Then OG is a single block, with defect group D = G maximal possible. (b) Let χ ∈ Irr(G) with χ(1)p = |G|p , then the corresponding block B of G has Irr(B) = {χ} and is called of defect zero. Here D = 1. For example, if p does not divide |G|, then every p-block of G is of defect zero. All other blocks contain at least two characters. For G = Sn , it is clear from the hook formula in Example 2.2 that χλ is of defect zero if and only if λ has no p-hook, that is, if and only if λ is a p-core. (c) The block B0 of G containing the trivial character 1G is called the principal block of G. It always has defect group D ∈ Sylp (G). With blocks now at our disposal, McKay’s Conjecture 2.1 can be naturally refined and generalised as follows (see [1]): Conjecture 2.5 (Alperin–McKay (1976)) Let B be a p-block of G with defect group D and Brauer correspondent b in NG (D). Then | Irr0 (B)| = | Irr0 (b)|, where Irr0 (B) := {χ ∈ Irr(B) | χ(1)p = |G : D|p }. The characters in Irr0 (B) are called characters of height zero. Let us point out that for blocks with defect group D ∈ Sylp (G) we have Irr0 (B) = Irr(B) ∩ Irrp (G), i.e., χ ∈ Irr(B) lies in Irr0 (B) if and only if χ ∈ Irrp (G). It follows that the Alperin–McKay conjecture implies the McKay conjecture, by just summing over all blocks of full defect. Again, the Alperin–McKay conjecture gives a local answer to a global question. It has been proved for all p-solvable groups by Okuyama–Wajima and Dade in 1980 [25, 71], for the symmetric groups, the alternating groups and their covering groups by Olsson [72] and Michler–Olsson [62]. In view of this conjecture it is of interest to know when it will provide information on all of Irr(B), that is, when all characters in Irr(B) are of height 0. This is the subject of another even older conjecture by Brauer [12]:

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Conjecture 2.6 (Brauer (1955)) Let B be a block with defect group D. Then Irr(B) = Irr0 (B)

⇐⇒

D is abelian.

A consequence of this so-called Brauer’s Height Zero Conjecture would be an easy criterion to decide from the character table of a finite group whether its Sylow p-subgroups are abelian: indeed, any Sylow p-subgroup D ∈ Sylp (G) is a defect group of the principal block B0 , and both Irr(B0 ) and Irr0 (B0 ) are encoded in the character table. Conjecture 2.6 has been proved for p-solvable groups by Gluck and Wolf [37], and for 2-blocks with defect group D ∈ Syl2 (G) much more recently by Navarro and Tiep [69] using, among other ingredients, Walter’s classification of groups with abelian Sylow 2-subgroup. Let us introduce a further fundamental conjecture in this subject. This purports to count irreducible characters in positive characteristic. For this let IBr(G) denote the set of irreducible p-Brauer characters of G; these are lifts to characteristic zero, constructed by Brauer, of the trace functions of irreducible representations G → GLn (Fp ). Again these are partitioned according to the p-blocks of G, so that IBr(G) = IBr(B1 )  . . .  IBr(Br ). A weight of G is a pair (Q, ψ) consisting of a p-subgroup Q ≤ G and an irreducible character ψ ∈ Irr(NG (Q)/Q) of defect zero. Clearly, G acts on its set of weights by conjugation. Any weight is naturally attached to a well-defined p-block of G. Then the Alperin Weight Conjecture [2] proposes: Conjecture 2.7 (Alperin (1986)) Let B be a block of G. Then | IBr(B)| = |{weights of G attached to B} ∼G |. So, | IBr(B)| should be determined locally, or more precisely this is the case whenever B is not of defect zero, since for blocks B of defect zero, with Irr(B) = {χ} (see Example 2.4(b)), the corresponding weight is just (1, χ). A proof of the Alperin Weight Conjecture 2.7 for solvable groups was given by Okuyama [70], for p-solvable groups by Isaacs and Navarro [44], for GLn (q) and Sn by Alperin and Fong [4], and for groups of Lie type when p is their defining prime by Cabanes [17]. For blocks with abelian defect groups (and hence in particular for groups with abelian Sylow p-subgroups), the weight conjecture has the following nice consequence: Theorem 2.8 (Alperin (1986)) Let B be a block with abelian defect groups satisfying the Alperin weight conjecture, and b its Brauer correspondent. Then: | Irr(B)| = | Irr(b)|

and

| IBr(B)| = | IBr(b)|.

Kn¨orr and Robinson [49] have given reformulations of Conjecture 2.7 in terms of chains of p-subgroups of G. They also showed the following connection between the conjectures introduced above: Theorem 2.9 (Kn¨ orr–Robinson (1989)) The following are equivalent for a prime p:

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(i) The Alperin–McKay Conjecture 2.5 holds for all p-blocks with abelian defect; (ii) the Alperin Weight Conjecture 2.7 holds for all p-blocks with abelian defect. While all of the above conjectures are open in general, they have been shown to hold for special types of defect groups. By results of Dade they hold whenever the defect group D is cyclic, and by Sambale [75] when D is metacyclic. Let us mention some further directions which we shall not go into here: several refinements of the above conjectures have been put forward, like the Isaacs–Navarro Conjecture 2.3 introduced above. Further, Dade’s conjecture [26] from 1992 simultaneously generalises the Alperin–McKay conjecture and the Alperin weight conjecture by making predictions on characters of arbitrary height. A recent conjecture of Eaton and Moreto [30] extends Brauer’s Height Zero Conjecture 2.6 to characters of the first positive height. 2.3

The reduction approach

In recent years a new approach for tackling the local-global counting conjectures has emerged: one tries to study a minimal counterexample by making use of the classification of the finite simple groups. A first such reduction in fact dates back quite a while [7]: Theorem 2.10 (Berger–Kn¨ orr (1988)) The “if ” direction of Brauer’s Height Zero Conjecture 2.6 holds if it holds for all blocks of all quasi-simple groups. Recall here that a finite group G is quasi-simple if G is perfect and moreover G/Z(G) is simple. It took 25 years until the necessary statement for quasi-simple groups could finally be verified, thus giving: Theorem 2.11 (Kessar–Malle (2013)) The “if ” direction of Brauer’s Height Zero Conjecture 2.6 holds. This result is the outcome of work of many mathematicians on determining all p-blocks of all quasi-simple groups, the case of groups of Lie type being by far the most challenging. Major contributions are due to Fong–Srinivasan [35], Brou´e– Malle–Michel [15], Cabanes–Enguehard [20], Blau–Ellers [8], Bonnaf´e–Rouquier [11] and Enguehard [32], before the final case, the so-called quasi-isolated blocks of exceptional groups of Lie type at bad primes was settled by Kessar and Malle [46]. About 15 years after Berger–Kn¨orr the issue of reductions of the long-standing conjectures was again taken up by Gabriel Navarro, which led to the following [43]: Theorem 2.12 (Isaacs–Malle–Navarro (2007)) The McKay Conjecture 2.1 holds for a prime p if all finite simple groups are McKay good for p. Here, the reduction is not as clean as for Brauer’s height zero conjecture. The condition of a simple group being McKay good is stronger and more complicated than just asking that it satisfies the McKay conjecture. We say that a simple group S of order divisible by p is McKay good at p if the following conditions hold, where

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G denotes a universal covering group of S (that is, G is maximal with respect to being quasi-simple with simple quotient S): Fix P ∈ Sylp (G). There exists a proper subgroup M < G of G with NG (P ) ≤ M such that (1) there is a bijection Ω : Irrp (G) → Irrp (M ), such that (2) Ω respects central characters, that is, if χ ∈ Irrp (G) lies above ν ∈ Irr(Z(G)), then so does Ω(χ) (note that Z(G) ≤ NG (P ) ≤ M ); (3) Ω is equivariant with respect to Aut(G)P = {α ∈ Aut(G) | α(P ) = P }; and (4) the Clifford theories in Aut(G)P above χ and Ω(χ) agree (for example, the corresponding 2-cocycles are the same). The last condition is often the most difficult to check, but it is satisfied automatically for example if Out(G) is cyclic; the latter property holds, for example, for sporadic simple groups, for alternating groups An with n = 6, but also for the exceptional groups of Lie type E8 (q). Let us give some indications on how such a reduction theorem might be arrived at. The first and crucial step is to generalise the desired assertion: Conjecture 2.13 (Relative McKay Conjecture) Let G be a finite group, L  G, P/L ∈ Sylp (G/L) and suppose that ν ∈ Irr(L) is P -invariant. Then | Irrp (G|ν)| = | Irrp (NG (P )|ν)|. The original McKay conjecture is recovered from this as the special case when L = 1, ν = 1. But, despite of seeming to be stronger, the relative version is much more accessible to an inductive approach. Indeed, Wolf [84] showed that Conjecture 2.13 holds for p-solvable groups. To prove the relative conjecture, let (G, L, ν) be a minimal counterexample with respect to |G/L|. Step 1: We may assume that ν is G-invariant: Let L ≤ T := Gν be the stabiliser of ν in G. By assumption P ≤ T , so |G : T | and N : N ∩ T | are prime to p. Clifford theory now yields bijections Irr(T |ν) → Irr(G|ν) and Irr(T ∩ N |ν) → Irr(N |ν) preserving the sets of p -degrees. Thus, if T < G then G cannot be a minimal counterexample. Step 2: We may assume L ≤ Z(G) is a cyclic p -group and ν is faithful: This uses the well-established theory of character triples: there exists a triple (G∗ , L∗ , ν ∗ ) with L∗ ≤ Z(G∗ ) cyclic, ν ∗ ∈ Irr(L∗ ) faithful and G/L ∼ = G∗ /L∗ ∗ ∗ such that the Clifford theories in G above ν and in G above ν agree (see e.g. [42, p. 186]). Step 3: We may assume that G/L has a unique minimal normal subgroup K/L, of order divisible by p: Let K/L be a minimal normal subgroup of G/L. Then |G/K| < |G/L|. Set

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M := NG (KP ). If M < G then we may conclude by induction. So we have M = G, and hence KP G. But then G/K is p-solvable. If G/L is p-solvable, then so is G, and we may conclude by the theorem of Wolf. Hence, G/L is not p-solvable but G/K is. From this it easily follows that K/L is the unique minimal normal subgroup. Step 4: We are done if the simple composition factors of K/L are McKay good for p: This is by far the most difficult part of the argument in [43], and we will not go into it here. A streamlined version of the arguments for this and more general reductions has been published by Sp¨ ath [82]. Since the publication of Theorem 2.12 all conjectures introduced above have been shown to reduce to properties of simple groups: 1. the Alperin–McKay Conjecture 2.5 holds if all simple groups are AMcK good (Sp¨ath [79]); 2. the Alperin Weight Conjecture 2.7 holds if all simple groups are AWC good (Navarro–Tiep [68], and Sp¨ ath [80] for the blockwise version); 3. the “only if” direction of Brauer’s Height Zero Conjecture 2.6 holds if all simple groups are AMcK good and moreover it holds for all quasi-simple groups (Navarro–Sp¨ ath [67]). The assertion on quasi-simple groups necessary for BHZ was subsequently shown by Kessar–Malle [48]. Moreover, for blocks B with abelian defect groups, Koshitani and Sp¨ath [50] show that being AWC good is implied by being AMcK good, if moreover the p-modular decomposition matrix of G is lower uni-triangular with respect to an Aut(G)-stable subset of characters. Thus the reductions have also uncovered some remarkably strong connections between the various conjectures. We will not endeavour to spell out the somewhat technical conditions for being good in the various cases, let us just say that they are similar to the one of being McKay good explained above. See [82], for example. Sp¨ath [81] has also succeeded in reducing Dade’s conjecture to a property of simple groups. So now all of the conjectures have been reduced to questions on finite simple groups, can we solve them? Well, it turns out that our knowledge on the representation theory of quasi-simple groups is not yet well-developed enough to really answer these questions. Roughly speaking, the alternating groups can be treated combinatorially (see also Example 3.7 for an illustration in symmetric groups), extending the aforementioned results of Olsson and Alperin–Fong to accommodate the stronger inductive conditions (see Denoncin [28]), the sporadic simple groups can be treated by ad hoc case-by-case methods (and this has been completed by various authors, except for the Alperin weight conjecture for the very largest sporadic groups, see e.g. An and Dietrich [6] and Breuer [14]). Similarly, the case of the finitely many exceptional covering groups of the simple groups of Lie type have been settled. Thus we are left with the by far biggest class of examples: the 16

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infinite families of simple groups of Lie type. Here, the case when p is the defining prime has been shown to hold by Sp¨ ath for all conjectures [78, 79, 80] building on previous work of Maslowski [60]. For the rest of these lectures we will concentrate on the McKay conjecture for groups of Lie type. First we need to understand the sets Irr(G) and Irr(NG (P )), or Irr(M ) for a suitable proper subgroup NG (P ) ≤ M < G. 2.4

McKay’s conjecture for GLn (q)

Let’s take a look at the case of G = GLn (q), q = pf a prime power. Here, the ordinary character table was determined by Green [39] in 1955. We need two ingredients. First let ⎛ ⎞  ∗ ... ∗  ⎜ ⎟ B = ⎝ ... ⎠ ≤ G 0 ∗ be the Borel subgroup of G consisting of upper triangular invertible matrices, and consider the induced character 1G B (the permutation character of G on the cosets of B). Theorem 2.14 (Green (1955)) The constituents of 1G B are in bijection with partitions λ  n such that  1G χλ (1) ρλq , B = λn

∈ Irr(GLn (q)) is the character labelled by λ, and χλ ∈ Irr(Sn ) is as in where Example 2.2. ρλq

Thus, the permutation character of G on B decomposes similarly to the regular character of Sn . This result is one reason why GLn (q) is sometimes called a “quantisation of Sn ”, or “Sn = GLn (1)”. The proof of Theorem 2.14 rests on the fact that the endomorphism algebra EndCG (1G B ) is an Iwahori–Hecke algebra H(Sn , q) at the parameter q, which by Tits’ deformation theorem is isomorphic to the complex group algebra CSn of Sn . The constituents ρλq , λ  n, of 1G B occurring in Theorem 2.14 were later called the unipotent characters of G. Now let s ∈ GLn (q) be a p -element; then s is diagonalisable over a finite extension of  Fq (it is a semisimple element of G). Its characteristic polynomial has the form ri=1 fini with  suitable irreducible polynomials fi ∈ Fq [X] of degrees di = deg(fi ) such that i ni di = n. Then CGLn (q) (s) ∼ = GLn1 (q d1 ) × · · · × GLnr (q dr ). i

Now for partitions λi  ni , 1 ≤ i ≤ r, and ρλqdi the corresponding unipotent 1

r

characters of the factors GLni (q di ), we have an irreducible character ρλqd1 ⊗· · ·⊗ρλqdr of CG (s). Let us write S for the set of all pairs (s, λ), where s ∈ GL is a  n (q) semisimple element up to conjugation with characteristic polynomial fini , and λ = (λ1 , . . . , λr )  (n1 , . . . , nr ) is an r-tuple of partitions.

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Theorem 2.15 (Green (1955)) There is a natural bijection S −→ Irr(GLn (q)),

(s, λ) → ρs,λ ,

such that ρs,λ (1) = | GLn (q) : CGLn (q) (s)|p ·

r 

i

ρλqdi (1).

i=1

{ρs,λ }

The sets E(G, s) := ⊆ Irr(G) are called Lusztig series. Observe that by its definition E(G, s) is in bijection with the set of r-tuples {(λ1 , . . . , λr )  (n1 , . . . , nr )} of partitions, which in turn parametrise the unipotent characters of CG (s) = GLn1 (q d1 ) × · · · × GLnr (q dr ). This is called the Jordan decomposition of the characters in Irr(GLn (q)). Thus, in order to verify for example the McKay conjecture we need to know the unipotent character degrees. These turn out to be given by a quantisation of the hook formula that we already saw for the character degrees of Sn in Example 2.2 ρλq (1) = q a(λ) 

[n]q ! h [ (h)]q

,

where the product runs again over all hooks h of λ. Here, [m]q := (q m − 1)/(q − 1) for m ≥ 1, [n]q ! := [1]q · · · [n]q , and a(λ) := i (i − 1)λi when λ = (λ1 ≥ λ2 ≥ . . .). There are two cases to discuss for the McKay conjecture: either the relevant prime equals p, or it is different from p, in which case we will call it . Let us first prove the following: Proposition 2.16 We have Irrp (GLn (q)) = {ρs,λ | λ = ((n1 ), . . . , (nr )), s ∈ GLn (q) semisimple}, so Irrp (GLn (q)) is in bijection with the semisimple conjugacy classes of GLn (q). Proof By the degree formula given above, p does not divide ρs,λ (1) if and only if   a(λi ) = 0, that is, if and only if all partitions λi are of the form λi = (ni ). Let us now consider the local side. Here ⎛ ⎞ ∗   1 ⎜ ⎟ P = ⎝ ... ⎠ ≤ G 0 1 is a Sylow p-subgroup of G, and NG (P ) = B = P.T with ⎛ ⎞ 0   ∗ ⎜ ⎟ T = ⎝ ... ⎠ ≤ G 0 ∗ an abelian subgroup (a so-called maximally split maximal torus of G). Thus Irrp (NG (P )) = Irrp (B) = Irrp (P.T ) = Irrp (P/P  .T ).

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n−1 , Maslowski [60] showed that Irrp (P/P  .T ) can be parametrised by F× q × (Fq ) and thus by the set of monic polynomials of degree n over Fq with non-vanishing constant coefficient, which are exactly the possible characteristic polynomials of semisimple elements in GLn (q). By Proposition 2.16 this establishes McKay’s conjecture for GLn (q) and the prime p. He also showed that the constructed bijection is equivariant with respect to AutP (G) (the relevant outer automorphisms are field automorphisms and the transpose-inverse automorphism when n > 2), and also compatible with respect to central characters. Now what about primes = p? Here we have the following observation, which is again immediate from the degree formula:

Proposition 2.17 Let = p. Then ρs,λ ∈ Irr (GLn (q)) if and only if s centralises i a Sylow -subgroup of GLn (q) and ρλqdi ∈ Irr (GLni (q di )) for all i = 1, . . . , r. That is, in order to prove the McKay conjecture in this case we are reduced to i understanding the unipotent characters ρλ , and for this, to determine when does divide a factorq m − 1. Let Φd denote the dth cyclotomic polynomial over Q, so that q m − 1 = d|m Φd (q). We also write d (q) for the order of q in F×  , that is, its order modulo . We have the following elementary criterion: Lemma 2.18 Let d = d (q). Then divides Φm (q) if and only if m ∈ {d, d , d 2 , . . . }. Then the degree formula implies: Corollary 2.19 Let > 2. For λ  n we have ρλ ∈ Irr (GLn (q)) if and only if λ has exactly w hooks of length d, where d = d (q) and n = wd + r with 0 ≤ r < d. Let’s now turn to the local situation. Proposition 2.20 Let d = d (q) and write n = wd + r with 0 ≤ r < d. The normaliser NG (P ) of a Sylow -subgroup P of G := GLn (q) is contained in



NG (GL1 (q d )w ) = (GL1 (q d ).Cd )  Sw × GLr (q) ∼ = GL1 (q d )w .G(d, 1, w) × GLr (q), where G(d, 1, n) = Cd  Sw is an imprimitive complex reflection group. The structure of a Sylow -normaliser is quite complicated in general, but by the Theorem Reduction

2.12 we can instead consider the intermediate group M := GL1 (q d )  G(d, 1, w) × GLr (q) which is much closer to being a finite reductive group like GLn (q) itself. Now taking together [54] and the result of Sp¨ath [77] show: Theorem 2.21 (Malle, Sp¨ ath (2010)) Let G = GLn (q) and M as above. There is a bijection Irr (G) → Irr (M ).

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The proof relies on combinatorial descriptions of the sets on both sides that coincide. This sketch shows how to find a McKay bijection in the case of GLn (q). A very similar statement also holds for the general unitary groups, but the proofs are different and more complicated. Now the general linear groups are in general not quasi-simple; the right groups to consider are the special linear groups SLn (q), where unfortunately the situation is much less transparent. Still, Cabanes and Sp¨ath [22] showed how to descend the bijection from Theorem 2.21 to SLn (q) and thus show that this group is McKay good for all . 2.5

Groups of Lie type

We now turn to general groups of Lie type. A finite group G is said to be of Lie type if G = GF , where G is a connected reductive group over an algebraic closure of a finite field with a Frobenius map F : G → G (we refer to [59] for an introduction to the structure theory of these groups). A subgroup H of G is said to be F -stable if F (h) ∈ H for all h ∈ H. If H ≤ G is F -stable, then HF is a finite group. A torus of G is a closed connected abelian subgroup of G consisting of semisimple elements. The group G acts on the set of tori of G; maximal tori form a single G-orbit. The group GF acts on the set of F -stable tori of G but there is in general more than one GF -orbit of F -stable maximal tori of G and F -fixed point subgroups of tori in different GF -classes have different orders. The GF -classes of maximal tori can be described using Weyl groups. Fix an F -stable maximal torus T of G and set W (T) = NG (T)/T, the Weyl group of T. The F -action on T induces an F -action on W (T). Elements w and w of W (T) are said to be F -conjugate if w = xwF (x)−1 for some x ∈ W (T). The GF -conjugacy classes of F -stable maximal tori are in bijection with the F -conjugacy classes of W which in turn can be described combinatorially. Example 2.22 Let G = GLn (Fq ) and F : G → G be the standard Frobenius map which sends every entry of a matrix in G to its q-th power. Then G = GF = GLn (q). Let T be the subgroup of G consisting of all diagonal matrices. Then T is an F -stable maximal torus of G. It is easy to see that NG (T) consists of all monomial matrices and W (T) ∼ = Sn . Further, since F fixes every permutation matrix the induced action of F on W (T) is trivial. So, the F -conjugacy classes of W (T) are simply the conjugacy classes of W (T). We obtain a bijection 1:1

{partitions of n} −→ {F -stable maximal tori of G}/GF . If λ = (λ1 , . . . , λs )  n corresponds to the GF -class of the maximal torus Tλ then |TFλ | = (q λ1 − 1) · · · (q λs − 1). In their seminal 1976 paper [27], Deligne and Lusztig showed how to construct ordinary GF -representations from the -adic cohomology spaces of certain algebraic

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varieties (now called Deligne–Lusztig varieties) on which GF acts. For each F stable maximal torus T of G, they constructed a pair of Z-linear maps G : Z Irr(TF ) → Z Irr(GF ), RT

∗ G RT

: Z Irr(GF ) → Z Irr(TF ).

G and ∗RG are called the Deligne–Lusztig induction and restriction The maps RT T maps respectively. These maps are adjoint to each other with respect to the standard scalar product on the space of class functions of GF , that is for each χ ∈ Irr(GF ), θ ∈ Irr(TF ), G G χ, RT (θ) = ∗RT (χ), θ .

For an F -stable maximal torus T of G and θ an irreducible character of TF , let E(GF |(T, θ)) be the subset of χ ∈ Irr(GF ) consisting of those χ such that G (θ) = χ, RT  0. The group GF acts by conjugation on the set of pairs (T, θ) where T is an F -stable maximal torus of G and θ is an irreducible character of TF and this action preserves the sets E(GF |(T, θ)), that is for all g ∈ GF , T an F -stable maximal torus of G and θ ∈ Irr(TF ), E(GF | g (T, θ)) = E(GF |(T, θ)). G (θ) as (T, θ) runs over all pairs of F -stable maximal tori The virtual characters RT T of G and irreducible characters θ of TF “trap” all irreducible characters of GF :

Theorem 2.23 (Deligne–Lusztig (1976))  E(GF |(T, θ)) Irr(GF ) = (T,θ)

as (T, θ) runs over the GF -conjugacy classes of pairs (T, θ) where T is an F -stable maximal torus of G and θ is an irreducible character of TF . 2.6

Characters of groups of Lie type

Let G be connected reductive with a Frobenius map F : G → G. If G is simple of simply connected type (like, for example, G = SLn ), then GF is, apart from a few exceptions, a finite quasi-simple group of Lie type. Moreover, all such groups, except for the Ree and Suzuki groups for which a slightly more general setup is needed, are obtained in this way. This turns out to be the right setting to study the character theory of the families of groups of Lie type. Recall that for T ≤ G an F -stable maximal torus, and θ ∈ Irr(TF ) there is G (θ). As for GL (q) the set of an associated virtual Deligne–Lusztig character RT n F irreducible characters of G = G can be partitioned into Lusztig series, as follows. Define a graph on Irr(G) by connecting two characters χ, χ ∈ Irr(G) if there exists G (θ) = 0 = χ , RG (θ) . The connected components a pair (T, θ) such that χ, RT T of this graph are the Lusztig series in Irr(G). This also defines an equivalence relation on the set of pairs (T, θ) which seems a bit mysterious. Lusztig has shown that the Lusztig series can instead also be parametrised by semisimple classes of a

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group G∗ closely related to G. It is obtained from the Langlands dual group G∗ of G as fixed points under a Frobenius map that we will also denote by F . Here, the Langlands dual G∗ has root datum obtained from that of G by exchanging character group and cocharacter group. For example, GL∗n = GLn , SL∗n = PGLn , Sp∗2n = SO2n+1 , E8∗ = E8 , . . . One usually writes E(G, s) ⊆ Irr(G) for the Lusztig series indexed by s ∈ G∗ . Example 2.24 Let G = GLn (q), s ∈ G∗ = GLn (q) semisimple. Let T ≤ CG (s) be an F -stable maximal torus. Then s ∈ T = TF corresponds to some θ ∈ Irr(T ) under the isomorphisms T ∼ = Irr(T ) induced by the duality between G and G∗ . Then  E(G, s) = E(GF |(T, θ)), T,θ

the union running over all such pairs (T, θ). In particular when s = 1 then all tori T contain s, and s corresponds to the trivial character 1T of T , so  E(G, 1) = E(GF |(T, 1T )) T

and these are the unipotent characters of G. Lusztig has shown that they are parametrised independently of q by suitable combinatorial data only depending on the complete root datum (the type) of (G, F ). For example, we had already seen that for G = GLn (q), the unipotent characters are parametrised by partitions of n, independently from q. As for GLn (q) there is a Jordan decomposition, which we state here only in a special situation, see [51]: Theorem 2.25 (Lusztig (1984)) Assume that s ∈ G∗ is such that CG∗ (s) is connected. Then there is a bijection Js : E(G, s) −→ E(CG∗ (s), 1), with χ(1) = |G∗ : CG∗ (s)|p · Js (χ)(1). Lusztig called this bijection the Jordan decomposition of irreducible characters. The assumption on s is satisfied for example for all semisimple elements in GLn (q), and more generally for all semisimple elements in groups G with connected centre, like, for example, PGLn or E8 . Example 2.26 Let s ∈ G∗ be such that CG∗ (s) = T∗ is a maximal torus of G∗ . The element s is then called regular. Regular semisimple elements are dense in G∗ , so this assumption is satisfied for “most” elements. In this case |E(G, s)| = 1, and χ(1) = |G∗ : T ∗ |p for {χ} = E(G, s).

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If CG∗ (s) is disconnected, the situation is considerably more complicated, but still Lusztig obtained an analogue of Jordan decomposition [52]. Example 2.27 Let G = SL2 (q) with q odd, so G∗ = PGL2 (q). The semisimple elements in G∗ are: the trivial element, two classes of elements of order 2 with disconnected centraliser (one lying inside PSL2 (q), one outside), and all other semisimple elements are regular with centraliser of order either q − 1 or q + 1. Letting s1 , s2 denote representatives of the two classes of involutions we thus have  Irr(G) = E(G, 1) ∪ E(G, s1 ) ∪ E(G, s2 ) ∪ E(G, s), s:s2 =1

where |E(G, 1)| = |E(G, si )| = 2, |E(G, s)| = 1; here s1 , s2 are representatives of the two classes of involutions. 2.7

Towards McKay’s conjecture for groups of Lie type

Again it is straightforward from the Jordan decomposition to classify the characters in Irr (G): Proposition 2.28 Let χ ∈ E(G, s). Then χ ∈ Irr (G) if and only if s centralises a Sylow -subgroup of G∗ and moreover Js (χ) ∈ E(CG∗ (s), 1) is contained in Irr (CG∗ (s)). So again our question is reduced to studying unipotent characters. Their degrees are given by polynomial expressions in the field size q, as we already saw for GLn (q) with the hook formula. It is combinatorially easy to determine the  -degrees from this for classical types; for exceptional types this is just a finite task. Now let’s turn again to the local picture. We set d = d (q), where we recall that d (q) denotes the order of modulo q. Assume for simplicity that = 2. We describe the picture for G of classical type, that is G = Gn (q) = Sp2n (q), SO2n+1 (q) or SO± 2n (q). First assume that d is odd and write n = ad + r with 0 ≤ r < d. Then there is a torus Td = GL1 (q d ) × · · · × GL1 (q d ) (a factors) of G such that NG (P ) ≤ NG (Td ) = Td .(C2d  Sw ) × Gr (q) contains the normaliser of a Sylow -subgroup P of Gn (q) (see [15, §3A]). ∼ n Example 2.29 Assume d = 1 and G = Gn (q) = SO− 2n (q). Then Td = Cq−1 is a maximally split torus, with NG (Td ) = Td .W , with W the Weyl group of G. If instead d = 2e is even, then the same type of result holds, we just have to replace the cyclic group GL1 (q d ) = Cqd −1 by the cyclic group GU1 (q e ) = Cqe +1 . Theorem 2.30 (Malle, Sp¨ ath (2010)) Let G be simple of simply connected type, = p a prime, and d = d (q). Then there exists a bijection Ω : Irr (G) → Irr (NG (Td )) with Ω(χ)(1) ≡ ±χ(1) (mod ) for all χ, unless one of • = 3, G = SL3 (q), SU3 (q), or G2 (q) with q ≡ 2, 4, 5, 7 (mod 9), or

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• = 2, G = Sp2n (q) with q ≡ 3, 5 (mod 8). This bijection can be chosen to preserve central characters. The proof is obtained by parametrising both sides by the same combinatorial data. For the listed exceptions NG (Td ) does not even contain a Sylow -subgroup; for example when G = Sp2 (q) ∼ = SL2 (q) with q ≡ 3, 5 (mod 8) a Sylow 2-subgroup is quaternion and thus cannot be contained in NG (Td ) which is an extension of a cyclic group of order q ± 1 by a group of order 2. Still the exceptions were shown to be McKay good [56]. Note that the group NG (Td ) only depends on d, but not on . Theorem 2.30 also gives the Isaacs– Navarro refinement from Conjecture 2.3. Now, what’s missing for proving McKay goodness? Equivariance and Clifford theory! Recall: for G quasi-simple of Lie type, Out(G) is made up of diagonal, graph and field automorphisms (see e.g.[59, Thm. 24.24]): 1. diagonal automorphisms are induced e.g. by the embedding of SLn (q) into GLn (q), or Sp2n (q) into CSp2n (q), 2. graph automorphisms come from the Dynkin diagram (e.g., the transposeinverse automorphism for SLn (q), n ≥ 3, or triality on D4 (q)), 3. field automorphisms come from the field Fq over which G is defined. Example 2.31 The worst case, in the sense that the structure of the outer automorphism group is most complicated, occurs for G = Spin+ 8 (q) with q ≡ 1 mod 2; here Out(G) = 22 .S3 .Cf , where q = pf . Nice cases (with small outer automorphism group) are, by contrast, G = E8 (q) or G = Sp2n (q) with q even; here Out(G) = Cf is cyclic. Theorem 2.32 (Cabanes–Sp¨ ath (2013)) Let S be simple of Lie type such that Out(S) is cyclic, then the bijection in Theorem 2.30 can be made equivariant. In particular, S is then McKay good. In general, we need to solve the following hard problem: Problem 2.33 For G quasi-simple of Lie type, determine the action of Aut(G) on Irr(G). Partial results are available: Lusztig determined the action of diagonal automorphisms: they leave χ ∈ E(G, s) invariant unless possibly when CG∗ (s) is disconnected. Also, the action of all automorphisms is known on Lusztig series where CG∗ (s) is connected. Example 2.34 (Lusztig) Consider the case of unipotent characters χ ∈ E(G, 1). If G is quasi-simple then any automorphism of G fixes all unipotent characters, unless G is of type D2n , or B2 , F4 in characteristic 2, or type G2 in characteristic 3 (see e.g. [56]).

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˜ be a regular embedding, that is G ˜ is a connected reductive group Let G → G with a connected center and the same derived subgroup as G. For example, the inclusion of SLn in GLn is a regular embedding. We assume that the Frobenius ˜ The acendomorphism F extends to a Frobenius morphism, also denoted F , of G. ˜ F on G is by inner-diagonal automorphisms. Denote by D the group ˜=G tion of G ˜ Then Sp¨ of graph and field automorphisms of G. ath [78] showed the following: Theorem 2.35 (Criterion of Sp¨ ath) Assume there is an Aut(G)P -equivariant ˜ : Irr (G) ˜ → Irr (M ˜ ) compatible with multiplication by Irr(G/G). ˜ bijection Ω If ˜ there is χ ∈ Irr (G|χ) • for every χ ˜ ∈ Irr (G) ˜ with ˜ χ  Dχ ˜  D)χ = G (G and χ extends to (G  D)χ , and • the analogous condition holds on the local side, then G/Z(G) is McKay good for . ˜ = GLn (q), then So, for G = SLn (q), for example, one uses the bijection for G has to check the stabiliser condition and finally prove extendibility. This leads to the following situation at the time of writing: S simple group is McKay good for all primes, unless possibly when S is of type Bn (q), (2)Dn (q), (2)E6 (q) or E7 (q). There is one prime for which more can be said: = 2. Theorem 2.36 (Malle–Sp¨ ath (2016)) Let G be quasi-simple of Lie type, not of type A, and χ ∈ Irr2 (G). Then there exists a linear character θ ∈ Irr(B) where B ≤ G is a Borel subgroup, such that χ is a constituent of IndG B (θ), unless some cases when G = Sp2n (q) with q ≡ 3 (mod 4). The proof, which is not too hard, uses Lusztig’s Jordan decomposition and the degree formulas. But now B = U.T , with T ≤ G a maximal torus, and U ≤ ker(θ), so in fact θ ∈ Irr(T ), and G B G IndG B (θ) = IndB (InflT (θ)) = RT (θ). To check the criterion, we need to know the action of Aut(G) on the constituents of G (θ) with T ≤ B. But the decomposition of RG (θ) is controlled by the Iwahori– RT T Hecke algebra of the relative Weyl group W (θ) = NG (T, θ)/T : G EndCG (RT (θ)) = H(W (θ), q) ∼ = CW (θ).

(As seen in the case of GLn (q) above). This does allow us to compute the action of Aut(G) on Irr2 (G); more considerations are needed on the local side, and extendibility has to be guaranteed. Theorem 2.37 (Malle–Sp¨ ath (2016)) The McKay Conjecture 2.1 holds for the prime p = 2.

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In order to generalise this to other primes, one needs to work with more general Levi subgroups: Let P ≤ G be an F -stable parabolic subgroup, and L ≤ P an F stable Levi complement, with finite groups of fixed points L = LF ≤ P = PF ≤ G. Then there is the functor of Harish-Chandra induction G RL : CL-mod → CG-mod,

P M → IndG P InflL (M ).

The special case L = T ≤ P = B was considered above. L (μ) Now λ ∈ Irr(L) is called cuspidal if it does not occur as constituent of RM for any proper Levi subgroup M < L, μ ∈ Irr(M ). Again the decomposition of G (λ), with λ cuspidal, is controlled by the Iwahori–Hecke algebra of a relative RL Weyl group (Howlett–Lehrer [40]). To apply the above argument, one needs to understand the action of automorphisms on cuspidal characters. Theorem 2.38 (Malle (2017)) Let G be quasi-simple of Lie type. Then the action of Aut(G) on the cuspidal characters of G lying in quasi-isolated series is known. This does, however, not yet solve the extension problem, and moreover the local situation also needs to be studied.

3

Blocks and characters of finite simple groups

As seen in the discussion around the McKay conjecture, in order to make a success of the reduction strategy for the local-global conjectures one needs very detailed knowledge both of the character theory of finite simple groups as well as of their p-local structure. For the block-wise versions of these conjectures we require this information at a yet finer level. A first step would be to obtain workable descriptions of block partitions of irreducible characters and the corresponding defect groups. The block distribution problem for finite (quasi and almost) simple groups falls naturally into four cases: • sporadic groups • alternating groups • finite groups of Lie type in describing characteristic • finite groups of Lie type in non-describing characteristic Of these the most difficult case is the last. We will discuss this case at some length. Block distributions in sporadic groups can be worked out through the ATLAS character tables. The third case, namely the blocks of finite groups of Lie type in defining characteristic is in some sense the easiest as there are very few blocks (see Example 3.8). In Example 3.7 we give a flavour of the first case by describing the block distribution for finite symmetric groups. 3.1

Local block theory

In order to get started we need to recall some foundational results from local block theory. As in Section 2.2 let O ≥ Zp be a large enough extension. Let k := O/P

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be the residue field of O and ¯ : O → k, α → α ¯ := α + P, the natural epimorphism. The block decomposition OG = B1 ⊕ . . . ⊕ Br induces the unique decomposition ¯r ¯1 ⊕ . . . ⊕ B kG = B into ideals of kG where for any element a =  a direct sum of minimal two-sided  α g of OG we denote a ¯ := α ¯ g g∈G g∈G g g. These decompositions correspond to unique decompositions 1OG = eB1 + . . . + eBr , 1kG = eB¯1 + . . . + eB¯r , of 1OG and of 1kG into a sum of central primitive idempotents, called block idempotents of kG. The expression for eB¯ is obtained by reducing coefficients modulo p. ¯i ↔ e ¯ between the sets of blocks of OG, blocks Thus we have bijections Bi ↔ B Bi ¯i , eB or e ¯ we mean a of kG and block idempotents of kG. By a defect group of B Bi i ¯i ), Irr(eB ) or Irr(e ¯ ). defect group of Bi . Similarly we may denote Irr(Bi ) by Irr(B Bi i Block idempotents can be read off the character table of G. If B is a block of G, then  χ(1)  eB = χ(x)x−1 , |G| x∈Gp

χ∈Irr(B)

is the sum of central idempotents of KG corresponding to the elements of Irr(B). Example 3.1 For G a finite group, Op (G) denotes the smallest normal subgroup of G with quotient a p-group and Op (G) denotes the largest normal p-subgroup of G. (a) For any block B of G, eB ∈ OOp (G). (b) For any block B of G and any normal subgroup N of G, eB ∈ OCG (N ). In particular, if CG (Op (G)) ≤ Op (G), then the principal block is the unique block of G. (c) If G = G1 × G2 is a direct product then the block idempotents of OG are of the form e1 e2 where ei is a block idempotent of OGi , i = 1, 2.  Let Q be a p-subgroup of G. For an element a = g∈G αg g of kG set BrQ (a) =



αg g ∈ kCG (Q).

g∈CG (Q)

The Brauer map BrQ : kG → kCG (Q), restricts to a multiplicative map on Z(kG).

a → BrQ (a),

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Theorem 3.2 (Brauer’s first main theorem) Let D be a p-subgroup of G. The map e → BrD (e) induces a bijection between block idempotents of kG with defect group D and block idempotents of kNG (D) with defect group D. If B is a p-block of G with defect group D, then BrD (eB¯ ) is the block idempotent of the Brauer correspondent of B in kNG (D). A G-Brauer pair (also known as subpair ) is a pair (Q, e) where Q ≤ G is a p-subgroup of G and e is a block idempotent of kCG (Q). We denote by P(G) the set of G-Brauer pairs and for a block B of G we denote by P(B) the subset of P(G) consisting of Brauer pairs (Q, e) such that BrQ (eB¯ )e = 0, the elements of P(B) are called B-Brauer pairs. It is easily seen that there is a partition P(G) = P(B1 )  . . .  P(Br ) where B1 , . . . , Br are the blocks of G. We would like to relate the block decomposition of Irr(G) with the block decomposition of P(G). Brauer’s second main theorem gives us a way of doing this. For x a p-element of G and χ ∈ Irr(G), we let dx χ : CG (x) → O be the function defined by  χ(xy) if y ∈ Gp , x d χ(y) = 0 if y ∈ / Gp  . Theorem 3.3 (Brauer’s second main theorem) Let B be a block of G, x ∈ G a p-element and C a block of CG (x). Suppose that there exists χ ∈ Irr(B), ψ ∈ Irr(C) such that  dx χ, ψ := χ(xy)ψ(y) = 0. y∈CG (s)p

Then (x , eC ) ∈ P(B). In particular, if dx χ = 0, then P(B) contains an element of the form (x , e). The assignment χ → dx χ extends by linearity to a map dx from the set of Ovalued class functions on G to the set of O-valued class functions on CG (x). The map dx is called the generalised decomposition map with respect to x. When we want to emphasise the underlying group G, the generalised decomposition map is denoted dx,G . The set P(B) has a nice description when B is the principal block. Theorem 3.4 (Brauer’s third main theorem) Let B0 be the principal block of G and let (Q, e) ∈ P(G). Then (Q, e) ∈ P(B0 ) if and only if e is the idempotent of the principal block of CG (Q). The set P(G) is a G-set via x

(Q, e) = ( x Q, x e),

for all x ∈ G, (Q, e) ∈ P(G)

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where x a := xax−1 for x ∈ G, a ∈ kG. The subset P(B) is G-invariant for B a block of G. In [3] Alperin and Brou´e endowed P(G) and P(B) with a G-poset structure. Let (Q, e), (R, f ) ∈ P(G). We say that (Q, e) is normal in (R, f ) and write (Q, e)  (R, f ) if Q  R, x (Q, e) = (Q, e) for all x ∈ R and BrR (e)f = 0. We say that (Q, e) ≤ (R, f ) if there exists a chain of normal inclusions (Q, e) =: (Q0 , e0 )  . . .  (Qn , en ) := (R, f ) in P(G) starting at (Q, e) and ending at (R, f ). Theorem 3.5 (Alperin–Brou´ e (1979)) (P(G), ≤) is a G-poset. For any (R, f ) ∈ P(G) and any Q ≤ R, there exists a unique block e of kCG (Q) such that (Q, e) ≤ (R, f ). The sets P(B) as B runs over the blocks of G are the connected components of (P(G), ≤). For a block B of G, (a) P(B) is G-invariant and (1, eB¯ ) is the unique minimal element of P(B). (b) G acts transitively on the set of maximal elements of P(B) and an element (D, d) of P(B) is maximal if and only if D is a defect group of B. The following is sometimes known as Brauer’s extended first main theorem. Theorem 3.6 (Recognition of maximal Brauer pairs) Let (Q, e) ∈ P(G). Then (Q, e) is maximal if and only if there exists θ ∈ Irr(e) such that • Z(Q) ≤ ker(θ), • as a character of CG (Q)/Z(Q), θ is of defect 0, and • NG (Q, e)/QCG (Q) is a p -group. Example 3.7 As discussed in Example 2.4, an irreducible character χλ of Sn lies in a p-block of defect zero if and only if λ is a p-core. By the Nakayama conjecture, posed in [64] and proved by Brauer and Robinson [13], given partitions λ, λ of n,  the corresponding characters χλ and χλ lie in the same p-block of Sn if and only if λ and λ have the same p-core. Puig [73] showed how the block distribution of irreducible characters matches up with the block distribution of P(Sn ). Let Q be a p-subgroup of Sn . Then n = m+pw, where Q fixes m points and moves pw points in the natural permutation representation of Sn , and CSn (Q) = Sm × CSpw (Q),

NSn (Q) = Sm × NSpw (Q).

The action of Q ≤ Spw is fixed-point free and it is not hard to show that this implies that CCSpw (Q) (Op (CSpw (Q))) ≤ Op (CSpw (Q)). By Example 3.1(b), the principal block is the unique block of CSpw (Q). Thus by Example 3.1(c) every Brauer pair with first component Q is of the form ef where e is a block idempotent of kSm and f is the principal block idempotent (in fact the identity element) of kCSpw (Q).

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Next, we describe the inclusion of Brauer pairs. This is a difficult and subtle step and is carried out inductively — a crucial ingredient is the Murnaghan–Nakayama rule which is an inductive combinatorial rule for calculating values of irreducible characters of symmetric groups. Let Q be a p-subgroup of Sn containing Q and suppose that Q moves pw points and fixes m points. Since Q ≤ Q , w ≥ w and m ≤ m. Let e f  be a block of CSn (Q ) with e a block of Sm and f  the principal block idempotent of kCSpw (Q ). Then one can show that (Q, ef ) ≤ (Q, e f  ) if and only if λ is obtained from μ by adding a sequence of p-hooks for χλ ∈ Irr(e), χμ ∈ Irr(e ). Finally, we describe the maximal pairs. Applying Theorem 3.6 in both directions, one sees that (Q, ef ) is a maximal Sn -Brauer pair if and only if (1, e) is a maximal Brauer pair for Sm and (Q, f ) is a maximal Spw -Brauer pair. By the inclusion rule described above, (1, e) is a maximal Brauer pair for Sm if and only if e is the block idempotent of a block of Sm of defect zero, that is, a block whose unique irreducible character is of the form χλ where λ is a p-core. Since f is a principal block idempotent, by Brauer’s third main theorem (Q, f ) is a maximal Spw -Brauer pair if and only if Q is a Sylow p-subgroup of Spw . Thus, the G-conjugacy classes of maximal Sn -Brauer pairs are in bijection with pairs (μ, w) where w is a nonnegative integer such that pw ≤ n and μ is a partition of n − pw which is a p-core. By Theorem 3.5 the G-conjugacy classes of maximal G-Brauer pairs are in one-toone correspondence with the blocks of G. Hence we obtain a bijection between the set of blocks of Sn and pairs as above; the character χλ ∈ Irr(Sn ) lies in the block indexed by the pair (μ, w) if and only if μ is the p-core of λ. If a block B is indexed by the pair (μ, w), then w is called the weight of B and μ is called the core of B. If B has weight w and core μ, then a Sylow p-subgroup P of Spw is a defect group of B and the Brauer correspondent of B in NSn (P ) = Sm × NSpw (P )

where μ  m

has the form Bμ Bw where Bμ is the block of Sm indexed by the pair (μ, 0) and Bw is the principal block of NSpw (P ). The irreducible characters in the Brauer correspondent are of the form χμ .η, where η ∈ Irr(Cw ). From this, it is easy to check that the map χμ .η → η is a height preserving bijection between the set of irreducible characters of the Brauer correspondent of B and the set of irreducible characters of Cw . Thus we obtain: (I) Given any non-negative integer w, there is a height preserving bijection between the irreducible characters of a Brauer correspondent of a weight w-block of a symmetric group and the principal block of NSpw (P ), where P ≤ Spw is a Sylow p-subgroup. In particular, there is a height preserving bijection between the irreducible characters of the Brauer correspondents of any two weight w blocks of (possibly different) symmetric groups.

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In [31] Enguehard showed that the global analogue of the above statement also holds, namely: (II) Given any non-negative integer w, there is a height preserving bijection between the irreducible characters of any two weight w blocks of (possibly different) symmetric groups. Thus the problem of checking a desired local-global statement for blocks of symmetric groups can often be reduced to checking it for a single block of any given weight w. Let us consider Brauer’s height zero conjecture (Conjecture 2.6) for p = 2. Since blocks with the same weight have isomorphic defect groups, in order to prove the height zero conjecture for blocks of symmetric groups it suffices to prove that it holds for the principal block B of S2w . The defect groups of B are the Sylow 2-subgroups of S2w . Hence B has abelian defect groups if and only if w = 1. On the other hand, Irr(B) = {χλ | λ  2w and λ has empty 2-core}. Since B is the principal block, Irr0 (B) = Irr(B) ∩ Irr2 (S2w ). Thus we are reduced to checking the following statement: w ≥ 2 if and only if there is λ  n such that λ has empty 2-core and 2 divides χλ (1). The backward implication is immediate as the only partitions of 2 are (2) and (1, 1). Now suppose that w ≥ 2. Then λ = (2w − 1, 1) has empty 2-core — we first remove successively w − 1 horizontal hooks of length 2 from the first part of the Young diagram, then remove the remaining vertical 2-hook. The hook length formula (see Example 2.2) easily yields that χλ has even degree. The local-local and global-global bijections described in (I) and (II) above are shadows of deeper categorical equivalences. It is quite easy to deduce from the above discussion that any two Brauer correspondents of blocks of symmetric groups with the same weight are Morita equivalent. Much harder is the analogous global version proven by Chuang and Rouquier [24]: any two p-blocks of symmetric groups of the same weight are derived equivalent. Example 3.8 Groups of Lie type in characteristic p have very few p-blocks. The main structural reason for this is the Borel–Tits theorem. As in Section 2.5, let G be a simple algebraic group over Fp with a Frobenius endomorphism F : G → G and let G = GF . Suppose that G is simply connected and Z(G) = 1. Then the Borel–Tits theorem [59, Thm. 26.5] implies that if Q is a non-trivial p-subgroup of G, then CNG (Q) (Op (NG (Q))) ≤ Op (NG (Q)). By Example 3.1(b) applied to NG (Q), the principal block is the unique block of NG (Q). Put another way, the identity element is the unique central idempotent of kNG (Q). Now suppose that f is a block idempotent of kCG (Q). Then it is easy to

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see that the sum f  of the distinct NG (Q)-conjugates of f is a central idempotent of kNG (Q). Hence f  is the identity element of kNG (Q). From this it follows that f = f  is the identity element of kCG (Q), and consequently f is the principal block of kCG (Q). In other words, the only G-Brauer pair with first component Q is the pair (Q, f ), where f is the principal block idempotent of kCG (Q). Now Brauer’s third main theorem (Theorem 3.4) gives that if (Q, e) ∈ P(G) with Q = 1, then (Q, e) ∈ P(B0 ), where B0 is the principal block of G. We conclude that the irreducible characters χ of G lying outside the principal block are all of defect zero. It turns out that there is only one character of defect zero, namely the the Steinberg character [41, Thm. 8.3]. Thus G has precisely two blocks: the principal block and the block containing the Steinberg character. If the assumption that Z(G) = 1 is dropped, then we obtain more blocks, but the extra blocks are in bijection with the non-trivial elements of Z(G). More precisely, we have the following [41, Thm. 8.3]: Suppose that G is simple and simply connected. The blocks of non-zero defect of G are in bijection with the elements of Z(G) and all have the Sylow p-subgroups of G as defect groups. There is exactly one block of defect zero, namely the block containing the Steinberg character of G. 3.2

Blocks of groups of Lie type in non-defining characteristic

We continue in the setting and notation of Section 2.5, so G is a connected reductive group over Fp with a Frobenius endomorphism F : G → G and G = GF . Let be a prime different from p. Our aim is to give a broad idea of how the -block partition of Irr(G) can be described in terms of Lusztig’s parametrisation of Irr(GF ). For notational simplicity for any F -stable subgroup H of G or of the dual group G∗ , we will denote by H the F -fixed point subgroup HF . For a character χ of G and x ∈ G an -element denote by dx,G χ : CG (x) → O the function dx χ as defined for Brauer’s second main theorem (see Section 3.1). A key starting point is the following result which relates generalised decomposition maps to Deligne–Lusztig induction and restriction. Let T be an F -stable ◦ (x), the connected maximal torus of G, T := TF , and let x ∈ T . Set H := CG component of the centraliser of x in G. The group H is again a connected reductive group which is F -stable and H := HF is a normal subgroup of CG (x) which may be proper (equality holds for example if CG (x) is itself connected). However, since x is an -element, the general structure theory of connected reductive groups gives that the index of H in CG (x) is a power of . By definition, for any character χ of G, dx,G χ is a function which takes zero values on -singular elements. Since H contains all -regular elements of CG (x), we may regard dx,G χ as a function from H and ∗RH are defined. H to O. Note that T is a maximal torus of H so RT T Theorem 3.9 ∗ H x,G RT (d χ)

G = dx,T (∗RT (χ))

for all χ ∈ Irr(G),

that is, Deligne–Lusztig restriction commutes with generalised decomposition maps.

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For an F -stable maximal torus T of G let Irr(T ) denote the subset of irreducible characters of T of  -order, that is Irr(T ) consists of those irreducible characters θ such that T ≤ ker(θ). Let E(G,  ) denote the subset of Irr(G) consisting of those G (θ), χ = χ such that RT  0 for some F -stable maximal torus T of G and some θ ∈ Irr(T ) . The following theorem illustrates how Brauer’s local block theory and the theory of Deligne–Lusztig characters come together. Theorem 3.10 Let T be an F -stable maximal torus of G, θ ∈ Irr(T ) . Suppose that CG (T ) = T. (∗) Then all elements of E(G|(T, θ)) lie in the same -block B of G. Further, if e is the block idempotent of kT = kCG (T ) containing θ, then (T , e) is a B-Brauer pair. The proof of the above theorem goes the following way (details may be found in [46, Props. 2.12, 2.13, 2.16]). By general structure theory the group CG (Q) is a reductive algebraic group for any Q ≤ T . For simplicity we will assume that CG (Q) is also connected. Step 1: Let χ ∈ E(G|(T, θ)). By the adjointness of Deligne–Lusztig induction G (χ). Another and restriction, θ is a constituent of the virtual character ∗RT G (θ) and key property of these maps is that since χ is a constituent of RT ∗ G θ ∈ Irr(T ) , all irreducible constituents of RT (χ) belong to Irr(T ) . Write ∗ G RT (χ)

= aθ θ +



with aθ , aτ ∈ Z, and aθ = 0.

aτ τ,

τ ∈Irr(T ) \{θ}

Let x ∈ T . Since T is an abelian group, it follows easily from the definition of generalised decomposition maps that if θ1 , θ2 ∈ Irr(T ) then dx,T θ1 , θ2 =

1 θ1 , θ2 . |T |

G (χ) gives Applying this to the above expression for ∗RT G (χ)), θ = dx,T (∗RT

1 aθ = 0. |T |

By the commutation property in Theorem 3.9, G H x,G H (χ)), θ = ∗RT (d χ), θ = dx,G χ, RT (θ) dx,T (∗RT

where H = CG (x) and where the second equality holds by adjointness. Now Brauer’s Second Main Theorem 3.3 implies that there exists a B-Brauer pair (x , f ) such that E(CG (x)|(T, θ)) ∩ Irr(f ) = 0.

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Step 2: Let {x1 , . . . , xm } be a generating set of T and let Qi = x1 , . . . , xi , 1 ≤ i ≤ m. Applying Step 1 repeatedly with G replaced by CG (Qi ), we obtain a sequence of inclusions of B-Brauer pairs (Q1 , f1 ) ≤ . . . ≤ (Qm , fm ) such that E(CG (Qi )|(T, θ)) ∩ Irr(fi ) = 0

for all i.

Since Qm = T , the hypothesis (*) implies that CG (Qm ) = T . Since θ is in Irr(e) as well as in Irr(fm ), fm = e, and the uniqueness of inclusion of Brauer pairs allows us to conclude. Example 3.11 (a) Let G = GLn , G = GLn (q). If divides q − 1, then the F stable maximal torus T of diagonal matrices of G satisfies the hypothesis (*) of G (θ) lie in the same Theorem 3.10. Thus, for any θ ∈ Irr(T ) , all constituents of RT

-block. By contrast, if does not divide q − 1, then TF has trivial -part. (b) Suppose that G is simple of classical type A, B, C, or D and = 2. For any F -stable maximal torus T of G, all elements of E(G|(T, 1)) lie in the principal 2block of G [18]. The key property is that T2 is non-trivial for all F -stable maximal tori T (of all F -stable Levi subgroups) of G. One applies Step (1) of the proof of Theorem 3.10 to some non-trivial x in T2 and then proceeds by induction on the dimension (as algebraic group) of CG (x). The hypothesis (*) of Theorem 3.10 does not hold often enough to obtain satisfactory control of block distribution of characters. The strategy to get around this is to replace F -stable maximal tori by a certain class of well behaved F -stable Levi subgroups. Let P ≤ G be a parabolic subgroup and let L ≤ P be a Levi complement. If L is F -stable, then we have a pair of mutually adjoint linear maps, called Lusztig induction and restriction, G RL : Z Irr(LF ) → Z Irr(GF ),

∗ G RL

: Z Irr(GF ) → Z Irr(LF ),

G and ∗RG . The construction enjoying many of the same properties as the maps RT T involves the parabolic subgroup P, and hence strictly speaking the notation for G should include P. However, we take the liberty of omitting this as in almost RL all situations it is known that the construction is independent of the choice of P. If P is also F -stable then Lusztig induction of an irreducible character χ of L is the same as Harish-Chandra induction of χ as considered in the previous section. For λ ∈ Irr(L) we let E(G|(L, λ)) denote the set of irreducible constituents of G (λ). We have an analogue of Theorem 3.10 which we state under an assumption RL on the prime being “large enough”. This assumption can be replaced by other conditions, e.g. |E(L,  ) ∩ Irr(e)| = 1.

Theorem 3.12 Suppose that ≥ 7. Let L be an F -stable Levi subgroup of G and let λ ∈ E(L,  ). Suppose that CG (Z(L) ) = L.

(∗∗)

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Then all elements of E(G|(L, λ)) lie in the same -block B of G. Further, if e is the block idempotent of kL containing θ, then (Z(L) , e) is a B-Brauer pair. The advantage of Theorem 3.12 over Theorem 3.10 is that there are many Levi G and ∗RG are subgroups satisfying Condition (**)—the disadvantage is that RL L G ∗ G harder to work with than RT and RT . It is known that every Levi subgroup of G is of the form L = CG (S) where S ≤ G is a (not necessarily maximal) torus S. Clearly, if S is F -stable then so is L. The class of Levi subgroups that is well adapted to Condition (**) are centralisers of particular F -stable tori which we now describe. To every F -stable torus S of G is associated a monic polynomial PS (x) with integer coefficients called the polynomial order of S such that m

|SF | = PS (q m )

for infinitely many integers m.

The polynomial order of S is uniquely defined and is a product of cyclotomic polynomials Φd (x), d ∈ N. If the polynomial order of S is a power of Φd (x) for a single integer d, then we say that S is a Φd -torus. If L is the centraliser in G of a Φd -torus, then L is said to be a d-split Levi subgroup of G. The following theorem of Cabanes and Enguehard which we state under some simplifying hypotheses shows that the class of d-split Levi subgroups (for a particular d) satisfies (**). Theorem 3.13 (Cabanes-Enguehard (1999)) Suppose that Z(G) is connected, [G, G] is simply connected and ≥ 7. Let d = d (q) be the order of q modulo

. Then every d-split Levi subgroup of G satisfies condition (**) of Theorem 3.12. Example 3.14 Let G = GLn (Fq ), G = GLn (Fq ). If L is an F -stable Levi subgroup of G, then L∼ = GLa1 (q m1 ) × · · · × GLar (q mr )  for some positive integers ai and mi , 1 ≤ i ≤ r, such that i ai mi = n. The group L is d-split if and only if mi = d for all 1 ≤ i ≤ r. 3.3

Lusztig series and Bonnaf´ e–Rouquier reduction

Another key feature of block theory in non-defining characteristic is that the subset E(G,  ) controls the -block distribution of irreducible characters. This is made precise in the following theorem. Note that E(G,  ) is the union of the Lusztig series E(G, s) as s runs over conjugacy classes of semisimple elements of  -order in the dual group G∗ . Theorem 3.15 (Hiss (1989), Brou´ e-Michel (1988)) Let B be an -block of G. There exists a semisimple  -element s of G∗ , unique up to conjugacy in G∗ such that Irr(B) ∩ E(G, s) = ∅. If t is a semisimple element of G∗ such that E(G, t) ∩ Irr(B) = ∅, then t is conjugate in G∗ to s.

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For a semisimple  -element s of G∗ , let E (G, s) be the union of Lusztig series E(G, t) where t runs over all semisimple elements of G∗ whose  -part is G∗ conjugate to s. The above theorem implies that E (G, s) is a union of -blocks of G. Thus, the -block distribution problem can be broken down as follows. For each (conjugacy class of) semi-simple  -element s of G∗ describe: (I) The -block distribution of E(G, s). (II) For each non-trivial semisimple -element t in CG∗ (s) describe the -block distribution of E(G, st). This approach is compatible with Theorem 3.12 since if s is a semisimple  -element of L∗ for some F -stable Levi subgroup L of G, and λ ∈ E(L, s), then all elements of E(G|(L, λ)) lie in E(G, s). The following powerful theorem of Bonnaf´e and Rouquier [11] allows for a dramatic shrinking of the magnitude of the problem. Theorem 3.16 (Bonnaf´ e–Rouquier (2003)) Let s ∈ G∗ be semisimple such ∗ that CG∗ (s) ≤ L for some F -stable Levi subgroup L∗ of G∗ . Then the product of

-block algebras in E (G, s) is Morita equivalent to the product of -block algebras in E(L, s). So, inductively, we only need to study the blocks in Lusztig series E (G, s) such that s is quasi-isolated in G, that is, such that CG∗ (s) is not contained in any proper Levi subgroup of G. The blocks in non-quasi-isolated series can be “recovered” from blocks of groups of Lie type where the underlying algebraic group is of smaller dimension than that of G. There is a price to be paid here: since Levi subgroups of a simple algebraic group are not simple, while working in the inductive set-up we cannot restrict ourselves to only considering simple algebraic groups. We need to take into account all Levi subgroups as well. Recently, Bonnaf´e, Dat and Rouquier [9] have given an improvement of Theorem 3.16 which in most situations reduces the set of semisimple elements that need to be considered even further, namely to isolated elements. These are elements s such that the connected component of CG∗ (s) is not contained in a proper Levi subgroup of G. 3.4

Unipotent blocks and d-Harish-Chandra theory

The best understood class of blocks are the unipotent blocks. These are the blocks in E (G, 1). By Theorem 3.15, the unipotent blocks are precisely the blocks which contain a unipotent character. In [15], Brou´e, Malle and Michel generalised the Harish-Chandra theory of Howlett and Lehrer to the context of d-split Levi subgroups (see the discussion before Theorem 2.38). This d-Harish-Chandra theory is an important ingredient in the solution of the block distribution problem. For d a positive integer, let Ud (G) denote the set of all pairs (L, λ) such that L is a d-split Levi subgroup of G and λ is an irreducible unipotent character of L. We

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regard G as a d-split Levi subgroup of itself, so (G, χ) ∈ Ud (G) for any irreducible unipotent character χ of G. The set Ud (G) is a G-set via g

(L, λ) = ( g L, g λ),

for g ∈ G, (L, λ) ∈ Ud (G).

There is also an inclusion relation on Ud (G) which is defined as follows: (L, λ) ≤ (M, μ)

M if L ≤ M and λ is a constituent of RL (μ).

The pair (M, μ) is said to be a unipotent d-cuspidal pair of G if there does not exist a unipotent d-cuspidal pair (L, λ) ≤ (M, μ) with L proper in M. For (L, λ) ∈ Ud (G) and M an F -stable Levi subgroup of G containing L, we denote by NM (L, λ) the stabiliser in M of the pair (L, λ) and we denote by WM (L, λ) the group NM (L, λ)/L, the relative Weyl group of (L, λ). Theorem 3.17 (Brou´ e–Malle–Michel (1993)) Let d be a positive integer. (a) (Ud (G), ≤) is a G-poset. The connected components of (Ud (G), ≤) are precisely the sets E(G|(L, λ)) as (L, λ) runs over a set of representatives of G-conjugacy classes of d-cuspidal pairs of G. (b) Let (L, λ) be a d-cuspidal pair of G and let M be a d-split Levi subgroup of G containing L. There exists an isometry ZE(M |(L, λ)) ∼ = Z Irr(WM (L, λ)) G with Ind G intertwining RM WM (L,λ) . W (L,λ)

The proof of Theorem 3.17 is on a case by case basis and relies heavily on the combinatorics associated to unipotent characters. An especially delicate point is the transitivity of ≤ since Lusztig induction sends characters to virtual characters. A by-product of part (b) of the theorem is an explicit description of d-cuspidal pairs (L, λ) and the sets E(G|(L, λ)). For classical groups, this description is in terms of the combinatorial “yoga” associated to partitions and symbols labelling unipotent characters and is in terms of tables for exceptional groups. The set E(G|(L, λ)) for a given d-cuspidal pair (L, λ) is called the d-HarishChandra series above (L, λ). Thus, for any d ≥ 1, Theorem 3.17 provides a partition of the set of unipotent characters into d-Harish Chandra series. It turns out that when d is the order of q modulo and provided that is sufficiently large the partition into d-Harish-Chandra series coincides with the block partition of E(G, 1), and is also closely linked with the -block partition of P(G). The following theorem, which makes this more precise, was proved by Cabanes and Enguehard [19]. For very large it is due to Brou´e, Malle and Michel [15]. The first assertion is covered by Theorem 3.12 and Theorem 3.13. Theorem 3.18 (Brou´ e–Malle–Michel (1993), (1994)) Suppose that ≥ 7 and let d = d (q).

Cabanes–Enguehard

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(a) For any unipotent d-cuspidal pair (L, λ) of G there exists a unique -block BG (L, λ) of G containing all elements of E(G|(L, λ)). (b) The map (L, λ) → BG (L, λ) induces a bijection between the G-classes of unipotent d-cuspidal pairs and the set of unipotent blocks of G. (c) Irr(BG (L, λ)) ∩ E(G, 1) = E(G|(L, λ)) for all unipotent d-cuspidal pairs (L, λ) of G. (d) Suppose that [G, G] is simply connected. The map (M, μ) → (Z(M ) , e(μ)) is an order reversing bijection from (Ud (G), ≤) onto a subset of (P(G), ≤) where e(μ) denotes the block idempotent of kCG (Z(M ) ) associated to μ. (e) There exists a maximal BG (L, λ)-Brauer pair (D, e) such that • (Z(L) , e(λ))  (D, e); • CD (Z(L) ) ≤ Z(L) ; and • D/Z(L) is isomorphic to a subgroup of WG (L, λ). Theorem 3.18 provides a complete solution to the block distribution] problem for unipotent characters. In other words, it completes Part (I) of the programme outlined in Section 3.3 for s = 1. In fact, Cabanes and Enguehard also give a solution to Part (II). We describe this briefly. For simplicity, we assume that Z(G) is connected. Let r be an -element of G∗ . The assumption that ≥ 7 implies that the centraliser of r in G∗ is a Levi subgroup of G∗ , necessarily F -stable. Duality between G and G∗ yields a corresponding F -stable Levi subgroup C(r) ≤ G and a linear character rˆ of C(r). By Lusztig’s parametrisation of characters, the elements of E(G, r), are precisely the characters G RC(r) (ˆ r ⊗ η),

η ∈ E(C(r), 1),

for some  ∈ {±1}. Cabanes and Enguehard show that to each unipotent d-cuspidal pair (L , λ ) of C(r) is associated a unipotent d-cuspidal pair (L, λ) of G such that [L, L] = [L , L ]

and



L ResL [L,L]F λ = Res[L ,L ]F λ.

G (ˆ r ⊗ η) belongs to the block B(L,λ) if and only if η lies in the d-Chandra Then RC(r) series of C(r) above a d-cuspidal pair (L , λ ) associated to (L, λ). The condition ≥ 7 in the above theorem can be replaced by the weaker condition: is odd, good for G and ≥ 5 if G has a simple component of type D4 which contributes the triality group 3D4 (q) to GF . In [32] Enguehard treated the unipotent -blocks for the remaining primes. One obtains a slightly weaker analogue of Theorem 3.18. The main difference is that the assignment in part (b) of the theorem, while still onto, is no longer one-to-one. In order to obtain a bijection one replaces the set of unipotent d-cuspidal pairs with a slightly smaller set, namely the set of unipotent d-cuspidal pairs with central -defect. Enguehard also does Part (II) of the problem but only in the case that the center of G is connected. For disconnected center groups the problem is still open.

100 3.5

Kessar, Malle: Local-global conjectures and blocks General blocks

There has also been a lot of work done to generalise the results of the previous section to non-unipotent blocks. There are two inter-connected approaches to this generalisation: (i) develop a non-unipotent d-Harish Chandra theory (ii) use Jordan decomposition to carry over unipotent d-Harish Chandra theory to the non-unipotent case. In [20], using a hybrid of the two approaches, Cabanes and Enguehard proved an analogue of Parts (a) and (b) of Theorem 3.18 for non-unipotent blocks as well as a weak analogue of (c) provided is odd, good for G and ≥ 5 if G has a simple component of type D4 . They also describe defect group structure. In [33], Enguehard showed that provided that ≥ 7 and the center of G is connected, then block distribution of irreducible characters is highly compatible with Jordan decomposition. In particular, for any semisimple  -element s of G∗ , there is a bijection B → Bs between the blocks of G in E (G, s) and the unipotent blocks of CG∗ (s) such that there is a height preserving bijection between the irreducible characters of B and those of Bs and such that B and Bs have isomorphic defect groups (in fact B and Bs have isomorphic Brauer categories). In the same paper, Enguehard also describes the blocks for classical groups when = 2. The paper [46] described the block distribution of E (G, s) for G simple of exceptional type and

a bad prime. Combining all of the previous results with the theorem of Bonnaf´e and Rouquier, a uniform parametrisation of blocks for all G such that G is simple was given in [47].

4

On the other conjectures; open problems

To end this survey, let us briefly comment on the status of the inductive conditions for the other local-global conjectures beyond the McKay conjecture and on some related open problems. The work on block parametrisation described in these sections has been enough to verify Brauer’s height zero conjecture for quasi-simple groups [46], [47]. For the remaining conjectures, we will need to do much more. For the immediate future, the two main open problems which need to be resolved for blocks of finite quasi-simple groups of Lie type are: Problem 4.1 Complete the description of non  -characters in -blocks of finite groups of Lie type for small (bad) primes . Problem 4.2 Describe the relationship between the -block distribution of P(G) and Theorem 3.18. For the Alperin–McKay Conjecture 2.5 we still do not have control over the global nor over the local situation in general. The hope is that we can prove a reduction of the necessary conditions to so-called quasi-isolated blocks, in the spirit of the Bonnaf´e–Rouquier Theorem 3.16: While this result gives some kind of reduction for the global situation, we are still missing an analogous local result.

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For the Alperin weight conjecture, the following cases have been dealt with so far, see [57] and [76]: Theorem 4.3 (Malle (2014), Schulte (2016)) The groups An , 2 G (q 2 ), 2F (q 2 ), 3D (q) and G (q) are AWC good for all primes . 2 4 4 2

2B

2 (q

2 ),

The proof requires the determination of all weights of all radical subgroups; for exceptional groups of larger rank that seems quite challenging at the moment. For classical types, it might be possible to use results of An [5]. Here we hope for 1. a generic description of weights in terms of d-tori and their normalisers 2. a Bonnaf´e–Rouquier type reduction to a few special situations. Another ingredient might be the following: Problem 4.4 Show that the -modular decomposition matrices of blocks of quasisimple groups of Lie type are unitriangular. This statement might follow, at least in good characteristic, from properties of generalised Gelfand–Graev characters. If this were true, one could make use of the result of Koshitani–Sp¨ath [50] mentioned before. The unitriangularity will be with respect to a suitable subset of Irr(B): A linearly independent subset X ⊆ Irr(B) is called a basic set for B if every Brauer character ϕ ∈ IBr(B) is an integral linear combination of the elements of X. So in particular |X| = | IBr(B)|. The following is folklore: Conjecture 4.5 Any -block of a quasi-simple group of Lie type has a “natural” basic set. Geck and Hiss [36] exhibited such a basic set when is good for the underlying algebraic group G and does not divide the order of Z(G): in this case E(G, s) is a basic set for the union of blocks E (G, s). This is yet another situation in which the theories of Brauer and of Lusztig fit together perfectly. It is known that this statement can no longer hold when either divides |Z(G)|, or when is bad for G. In some cases, replacements have been found, but this is still open in general. Acknowledgement: We thank Britta Sp¨ ath and Yanjun Liu for their pertinent comments on a previous version. References [1] J. L. Alperin, The main problem of block theory. In: Proceedings of the Conference on Finite Groups, Univ. Utah, Park City, Utah (1975), Academic Press, New York, 1976, 341–356. [2] J. L. Alperin, Weights for finite groups. In: The Arcata Conference on Representations of Finite Groups, Arcata, Calif. (1986), Proc. Sympos. Pure Math. 47, Part 1, Amer. Math. Soc., Providence, 1987, 369–379.

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[3] J. L. Alperin and M. Brou´e, Local methods in block theory. Ann. of Math. 110 (1979), 143–157. [4] J. L. Alperin and P. Fong, Weights for symmetric and general linear groups. J. Algebra 131 (1990), 2–22. [5] J. An, Weights for classical groups. Trans. Amer. Math. Soc. 342 (1994), 1–42. [6] J. An and H. Dietrich, The AWC-goodness and essential rank of sporadic simple groups. J. Algebra 356 (2012), 325–354. [7] T. R. Berger and R. Kn¨orr, On Brauer’s height 0 conjecture. Nagoya Math. J. 109 (1988), 109–116. [8] H. Blau and H. Ellers, Brauer’s height zero conjecture for central quotients of special linear and special unitary groups. J. Algebra 212 (1999), 591–612. [9] C. Bonnaf´e, J. F. Dat and R. Rouquier, Derived categories and Deligne–Lusztig varieties II. Ann. of Math. (2) 185 (2017), 609–676. [10] C. Bonnaf´e and J. Michel, Computational proof of the Mackey formula for q > 2. J. Algebra 327 (2011), 506–526. [11] C. Bonnaf´e and R. Rouquier, Cat´egories d´eriv´ees et vari´et´es de Deligne–Lusztig. Publ. ´ Math. Inst. Hautes Etudes Sci. 97 (2003), 1–59. [12] R. Brauer, Number theoretical investigations on groups of finite order. In: Proceedings of the International Symposium on Algebraic Number Theory, Tokyo and Nikko (1955), Science Council of Japan, Tokyo, 1956, 55–62. [13] R. Brauer and G. de B. Robinson, On a conjecture by Nakayama. Trans. Roy. Soc. Canada. Sect III (3) 41 (1947), 11–25. [14] T. Breuer, Computations for some simple groups. At: http://www.math.rwth-aachen.de/∼Thomas.Breuer/ctblocks/doc/overview.html [15] M. Brou´e, G. Malle and J. Michel, Generic blocks of finite reductive groups. Ast´erisque 212 (1993), 7–92. [16] O. Brunat and R. Nath, The Navarro conjecture for the alternating groups. arXiv:1803.01423. [17] M. Cabanes, Brauer morphism between modular Hecke algebras. J. Algebra 115 (1988), 1–31. [18] M. Cabanes and M. Enguehard, Unipotent blocks of finite reductive groups of a given type. Math. Z. 213 (1993), 479–490. [19] M. Cabanes and M. Enguehard, On unipotent blocks and their ordinary characters. Invent. Math. 117 (1994), 149–164. [20] M. Cabanes and M. Enguehard, On blocks of finite reductive groups and twisted induction. Adv. Math. 145 (1999), 189–229. [21] M. Cabanes and M. Enguehard, Representation Theory of Finite Reductive Groups. Cambridge University Press, Cambridge, 2004. [22] M. Cabanes and B. Sp¨ ath, Equivariant character correspondences and inductive McKay condition for type A. J. reine angew. Math. 728 (2017), 153–194. [23] R. Carter, Finite Groups of Lie type: Conjugacy Classes and Complex Characters. Wiley, Chichester, 1985. [24] J. Chuang and R. Rouquier, Derived equivalences for symmetric groups and sl2 categorification. Ann. of Math. (2) 167 (2008), 245–298. [25] E. C. Dade, A correspondence of characters. In: The Santa Cruz Conference on Finite Groups, Univ. California, Santa Cruz, Calif. (1979), Amer. Math. Soc., Providence, R.I., 1980, 401–403. [26] E. C. Dade, Counting characters in blocks. I. Invent. Math. 109 (1992), 187–210; II. J. reine angew. Math. 448 (1994), 97–190. [27] P. Deligne and G. Lusztig, Representations of reductive groups over finite fields. Ann. of Math. 103 (1976), 103–161.

Kessar, Malle: Local-global conjectures and blocks

103

[28] D. Denoncin, Inductive AM condition for the alternating groups in characteristic 2. J. Algebra 404 (2014), 1–17. [29] F. Digne and J. Michel, Representations of Finite Groups of Lie Type. Volume 21 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1991. [30] C. Eaton and A. Moret´o, Extending Brauer’s height zero conjecture to blocks with nonabelian defect groups. Int. Math. Res. Notices 20 (2014), 5561–5581. [31] M. Enguehard, Isom´etries parfaites entre blocs de groupes sym´etriques. Ast´erisque 181-182 (1990), 157–171. [32] M. Enguehard, Sur les l-blocs unipotents des groupes r´eductifs finis quand l est mauvais. J. Algebra 230 (2000), 334–377. [33] M. Enguehard, Vers une d´ecomposition de Jordan des blocs des groupes r´eductifs finis. J. Algebra 319 (2008), 1035–1115. [34] P. Fong, The Isaacs–Navarro conjecture for symmetric groups. J. Algebra 260 (2003), 154–161. [35] P. Fong and B. Srinivasan, The blocks of finite classical groups. J. reine angew. Math. 396 (1989), 122–191. [36] M. Geck and G. Hiss, Basic sets of Brauer characters of finite groups of Lie type. J. reine angew. Math. 418 (1991), 173–188. [37] D. Gluck and T. R. Wolf, Brauer’s height conjecture for p-solvable groups. Trans. Amer. Math. Soc. 282 (1984), 137–152. [38] D. Gorenstein, R. Lyons and R. Solomon, The Classification of the Finite Simple Groups. American Mathematical Society, 40, Number 3. Providence, RI, 1998. [39] J. A. Green, The characters of the finite general linear groups. Trans. Amer. Math. Soc. 80 (1955), 402–447. [40] R. B. Howlett and G. I. Lehrer, Induced cuspidal representations and generalised Hecke rings. Invent. Math. 58 (1980), 37–64. [41] J. E. Humphreys, Modular Representations of Finite Groups of Lie Type. London Mathematical Society Lecture Note Series, 326, Cambridge University Press, Cambridge, 2006. [42] I. M. Isaacs, Characters of solvable and symplectic groups. Amer. J. Math. 95 (1973), 594–635. [43] I. M. Isaacs, G. Malle and G. Navarro, A reduction theorem for the McKay conjecture. Invent. Math. 170 (2007), 33–101. [44] I. M. Isaacs and G. Navarro, Weights and vertices for characters of π-separable groups. J. Algebra 177 (1995), 339–366. [45] I. M. Isaacs and G. Navarro, New refinements of the McKay conjecture for arbitrary finite groups. Ann. of Math. (2) 156 (2002), 333–344. [46] R. Kessar and G. Malle, Quasi-isolated blocks and Brauer’s height zero conjecture. Ann. of Math. (2) 178 (2013), 321–384. [47] R. Kessar and G. Malle, Lusztig induction and -blocks of finite reductive groups. Pacific J. Math. 279 (2015), 267–296. [48] R. Kessar and G. Malle, Brauer’s height zero conjecture for quasi-simple groups. J. Algebra 475 (2017), 43–60. [49] R. Kn¨orr and G. R. Robinson, Some remarks on a conjecture of Alperin. J. London Math. Soc. (2) 39 (1989), 48–60. [50] S. Koshitani and B. Sp¨ ath, The inductive Alperin-McKay and blockwise Alperin weight conditions for blocks with cyclic defect groups and odd primes. J. Group Theory 19 (2016), 777–813. [51] G. Lusztig, Characters of Reductive Groups over a Finite Field. Annals of Mathematics Studies, 107. Princeton University Press, Princeton, NJ, 1984.

104

Kessar, Malle: Local-global conjectures and blocks

[52] G. Lusztig, On the representations of reductive groups with disconnected centre. Ast´erisque 168 (1988), 157–166. [53] I. G. Macdonald, On the degrees of the irreducible representations of symmetric groups. Bull. London Math. Soc. 3 (1971), 189–192. [54] G. Malle, Height 0 characters of finite groups of Lie type. Represent. Theory 11 (2007), 192–220. [55] G. Malle, The inductive McKay condition for simple groups not of Lie type. Comm. Algebra 36 (2008), 455–463. [56] G. Malle, Extensions of unipotent characters and the inductive McKay condition. J. Algebra 320 (2008), 2963–2980. [57] G. Malle, On the inductive Alperin–McKay and Alperin weight conjecture for groups with abelian Sylow subgroups. J. Algebra 397 (2014), 190–208. [58] G. Malle and B. Sp¨ ath, Characters of odd degree. Ann. of Math. (2) 184 (2016), 869–908. [59] G. Malle and D. Testerman, Linear Algebraic Groups and Finite Groups of Lie Type. Cambridge Studies in Advanced Mathematics, 133, Cambridge University Press, Cambridge, 2011. [60] J. Maslowski, Equivariant Bijections in Groups of Lie Type. Dissertation, TU Kaiserslautern, 2010. [61] J. McKay, Irreducible representations of odd degree. J. Algebra 20 (1972), 416–418. [62] G. O. Michler and J. B. Olsson, The Alperin-McKay conjecture holds in the covering groups of symmetric and alternating groups, p = 2. J. reine angew. Math. 405 (1990), 78–111. [63] H. Nagao and Y. Tsushima, Representations of Finite Groups. Academic Press, San Diego, 1987. [64] T. Nakayama, On some modular properties of irreducible representations of a symmetric group I. Jap. J. Math. 18 (1941), 89–108. [65] R. Nath, The Isaacs–Navarro conjecture for the alternating groups. J. Algebra 321 (2009), 1632–1642. [66] G. Navarro, The McKay conjecture with Galois automorphisms. Ann. of Math. (2) 160 (2004), 1129–1140. [67] G. Navarro and B. Sp¨ath, On Brauer’s height zero conjecture. J. Eur. Math. Soc. 16 (2014), 695–747. [68] G. Navarro and P. H. Tiep, A reduction theorem for the Alperin weight conjecture. Invent. Math. 184 (2011), 529–565. [69] G. Navarro and P. H. Tiep, Brauer’s height zero conjecture for the 2-blocks of maximal defect. J. reine angew. Math. 669 (2012), 225–247. [70] T. Okuyama, Module correspondence in finite groups. Hokkaido Math. J. 10 (1981), 299–318. [71] T. Okuyama and M. Wajima, Character correspondence and p-blocks of p-solvable groups. Osaka J. Math. 17 (1980), 801–806. [72] J. B. Olsson, McKay numbers and heights of characters. Math. Scand. 38 (1976), 25–42. [73] L. Puig, The Nakayama conjecture and the Brauer pairs. In: S´eminaire sur les groupes finis, Tome III Publ. Math. Univ. Paris VII, 25 Paris (1986), 171–189. [74] L. Ruhstorfer, The Navarro refinement of the McKay conjecture for finite groups of Lie type in defining characteristic. Preprint, arXiv:1703.09006. [75] B. Sambale, Blocks of Finite Groups and Their Invariants. Lecture Notes in Mathematics, 2127. Springer, Cham, 2014. [76] E. Schulte, The inductive blockwise Alperin weight condition for G2 (q) and 3D4 (q). J. Algebra 466 (2016), 314–369.

Kessar, Malle: Local-global conjectures and blocks

105

[77] B. Sp¨ ath, Sylow d-tori of classical groups and the McKay conjecture. I. J. Algebra 323 (2010), 2469–2493; II. J. Algebra 323 (2010), 2494–2509. [78] B. Sp¨ ath, Inductive McKay condition in defining characteristic. Bull. Lond. Math. Soc. 44 (2012), 426–438. [79] B. Sp¨ ath, A reduction theorem for the Alperin–McKay conjecture. J. reine angew. Math. 680 (2013), 153–189. [80] B. Sp¨ ath, A reduction theorem for the blockwise Alperin weight conjecture. J. Group Theory 16 (2013), 159–220. [81] B. Sp¨ ath, A reduction theorem for Dade’s projective conjecture. J. Eur. Math. Soc. 19 (2017), 1071–1126. [82] B. Sp¨ath, Inductive conditions for counting conjectures via character triples. In: Representation Theory — Current Trends and Perspectives, Bad Honnef (2015), Eur. Math. Soc., Z¨ urich, 2017, 665–680. [83] A. Turull, Character correspondences in solvable groups. J. Algebra 295 (2006), 157– 178. [84] T. Wolf, Variations on McKay’s character degree conjecture. J. Algebra 135 (1990), 123–138.

A SURVEY ON SOME METHODS OF GENERATING FINITE SIMPLE GROUPS AYOUB B. M. BASHEER∗ and JAMSHID MOORI† ∗

Department of Mathematical Sciences, North-West University, P Bag X2046, Mmabatho 2735, South Africa Email: [email protected]

† Department of Mathematical Sciences, North-West University, P Bag X2046, Mmabatho 2735, South Africa Email: [email protected]

Abstract A finite group can be generated in many different ways. In this paper we consider a few methods of generating finite simple groups and in particular we focus on those of interest to the authors especially the second author and his research group. These methods are concerned with ranks of conjugacy classes of elements, (p, q, r)-, nXcomplementary generation and exact spread of finite non-abelian simple groups. We also give some examples of results that were established by the authors on generation of some finite non-abelian simple groups.

1

Introduction

Generation of finite groups by suitable subsets is of great interest and has many applications to groups and their representations. For example, the computations of the genus of simple groups can be reduced to the generation of the relevant simple groups (see Woldar [81] for details). Also Di Martino et al. [41] established a useful connection between generation of groups by conjugate elements and the existence of elements representable by almost cyclic matrices. Their motivation was to study irreducible projective representations of the sporadic simple groups. In view of applications, it is often important to exhibit generating pairs of some special kind, such as • generators carrying a geometric meaning, • generators of some prescribed order, • generators that offer an economical presentation of the group. The classification of finite simple groups is involved heavily and plays a pivotal role in most general results on the generation of finite groups. The study of generating sets in finite groups has a rich history, with numerous applications. We recall that a finite group is said to be 2-generated if it can be generated by two elements. Now let G be a finite non-abelian simple group. It is well known that G can be generated by two elements. This has been known for a long time, since Miller [64] in 1901, for the case of the alternating groups. In 1962 the result was

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extended by Steinberg [73] to the groups of Lie type, where he gave a unified treatment for the 2-generation of the Chevalley and the Twisted groups. Before this the 2-generation of certain families of groups of Lie type were known (e.g., P SL(n, F) and Sp(2n, F)). In 1984, Aschbacher and Guralnick [7] completed the problem of determining which of the finite simple groups are 2-generated by analyzing the sporadic groups that had not already been settled by other authors. They showed that any sporadic simple group can be generated by an involution and another suitable element. A 2017 paper by King [60] gave a refinement where it was shown that every finite non-abelian simple group is generated by an involution and an element of prime order. Turning to the maximal subgroups of finite simple groups, Burness et. al. [25] showed that the minimal number of generators of any maximal subgroup of a non-abelian finite simple group is 4 or less and that this bound is best possible. In general the topic of generation of finite simple groups is fairly rich and bigger than can be covered in few pages of a paper of the proceedings of this conference. Indeed this can be seen from the large list of references, which is yet not the full list of all the work done on this topic. In this article we cover some of the classical and recent results on the generation of finite simple groups by paying attention to some specific methods of generation. The following are a few of the problems concerning the generation of finite simple groups that may be found in the literature. • In his PhD Thesis [75], Ward considered the problem of generating a nonabelian finite simple group by a set of conjugate involutions whose product is the identity. More specifically, he considers the problem of which groups have the property of being generated by 5 conjugate involutions whose product is the identity (he referred to this as groups having Property 1) and which simple groups have the property that they can be generated by 3 conjugate involutions, a, b and c such that a and b commute and ab is also conjugate to a, b and c (he referred to this as groups having Property 2). Property 2 is indeed a stronger property than Property 1, that is if G has Property 2, then it has Property 1. • A group is said to be 32 -generated if every non-trivial element is contained in a generating pair. Guralnick and Kantor [54] showed that every finite simple group is 32 -generated. In [23], Breuer et. al., conjectured that any finite group is 32 -generated if and only if every proper quotient is cyclic and recent work of Guralnick [53] reduces this conjecture to almost simple groups. A 2017 paper by S. Harper [57] extended the results to almost simple symplectic and odd-dimensional orthogonal groups. • A generating set X ⊂ G is said to be irredundant or independent if no proper subset of X generates G. We write m(G) for the largest size of such a set. In [77] Whiston found that m(Sn ) = n−1, a result later generalized by Cameron and Cara in [27] by classifying the independent generating sets of Sn . In his thesis [78] Whiston also proved similar results for the L2 (q) (published in [79]) and the Suzuki groups. Later in another thesis [59] Keen proved similar results for the groups SO3 (q), SU3 (q) and SL3 (q) and in another thesis [24]

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Basheer, Moori: On the generation of finite simple groups Brooks showed that m(M11 ) = 5 and m(M12 ) = 6. • Guralnick et. al. [55, 54] were interested in probabilistic random generation of a finite simple group using elements of a fixed conjugacy class of the group. In fact Burness, Guralnick, Kantor, Liebeck, Saxl and Shalev have a pioneering role in the problem of the probabilistic random generation.

In addition to the above, there are many other problems concerning generating sets of groups. In this paper we restrict ourselves to the areas of generation of finite non-abelian simple groups that are of interest to the authors, especially the second author and his research group including postgraduate students and postdoctoral fellows. To be more precise, we will focus on the ranks of conjugacy classes of elements, (p, q, r)-generation, nX-complementary generation and exact spread. The common technique used in the first three methods is the structure constant method and thus, in the next section, we give a brief description for this method together with some results for generation and non-generation of finite non-abelian simple groups.

2

Preliminaries

Let G be a finite group and C1 , · · · , Ck be k ≥ 3 (not necessarily distinct) conjugacy classes of G with g1 , · · · , gk being representatives for these classes respectively. For a fixed representative gk ∈ Ck and for gi ∈ Ci , 1 ≤ i ≤ k − 1, denote by ΔG = ΔG (C1 , C2 , · · · , Ck ) the number of distinct (k − 1)-tuples (g1 , g2 , · · · , gk−1 ) such that g1 g2 · · · gk−1 = gk . This number is known as a class algebra constant or structure constant. With Irr(G) = {χ1 , χ2 , · · · , χr } being the set of complex irreducible characters of G, the number ΔG is easily calculated from the character table of G through the formula (see for example [62]) k−1 

ΔG (C1 , C2 , · · · , Ck ) =

|Ci |

i=1

|G|

r  χi (g1 )χi (g2 ) · · · χi (gk−1 )χi (gk ) i=1

(χi (1G ))k−2

.

(1)

Also for a fixed gk ∈ Ck we denote by Δ∗G (C1 , C2 , · · · , Ck ) the number of distinct (k − 1)-tuples (g1 , g2 , · · · , gk−1 ) ∈ C1 × C2 × · · · × Ck−1 satisfying g1 g2 · · · gk−1 = gk

and

g1 , g2 , · · · , gk−1 = G.

(2)

Definition 2.1 If Δ∗G (C1 , · · · , Ck ) > 0, the group G is said to be (C1 , C2 , · · · , Ck )generated. Furthermore if H ≤ G is any subgroup containing a fixed element gk ∈ Ck , we let ΣH (C1 , · · · , Ck ) be the total number of distinct (k−1)-tuples (g1 , g2 , · · · , gk−1 ) such that g1 g2 · · · gk−1 = gk and g1 , g2 , · · · , gk−1 ≤ H. The value of ΣH (C1 , C2 , · · · , Ck ) can be obtained as a sum of the structure constants ΔH (c1 , c2 , · · · , ck ) of H conjugacy classes c1 , c2 , · · · , ck such that ci ⊆ H Ci .

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Theorem 2.2 Let G be a finite group and H be a subgroup of G containing a fixed element g such that gcd(o(g), [NG (H):H]) = 1. Then the number h(g, H) of conjugates of H containing g is χH (g), where χH is the permutation character of G with action on the conjugates of H. In particular h(g, H) =

m  i=1

|CG (g)| , |CNG (H) (xi )|

where x1 , x2 , · · · , xm are representatives of the NG (H)-conjugacy classes fused to the G-class of g. Proof See for example [46, 44].



The above number h(g, H) is useful in giving a lower bound for Δ∗G (C1 , · · · , Ck ), namely Δ∗G (C1 , C2 , · · · , Ck ) ≥ ΘG (C1 , C2 , · · · , Ck ), where  h(gk , H)ΣH (C1 , · · · , Ck ), (3) ΘG (C1 , · · · , Ck ) = ΔG (C1 , · · · , Ck ) − gk is a representative of the class Ck and the sum is taken over all the representatives H of G-conjugacy classes of maximal subgroups containing elements of all the classes C1 , C2 , · · · , Ck . Remark 2.3 Since we have all the maximal subgroups of the sporadic simple groups (except for G = M the Monster group), it is possible to build a small subroutine in GAP [51] or in Magma [19] to compute the values of ΘG = ΘG (C1 , · · · , Ck ) for any collection of conjugacy classes of a sporadic simple group. Remark 2.4 If ΘG > 0 then certainly G is (C1 , C2 , · · · , Ck )-generated. We now quote some results for establishing if a given set of elements of a group generate it. These results are also important in determining the ranks and (p, q, r)generation of the finite simple groups. By nX we mean a conjugacy class X of G, where elements of X are of order n. Proofs for the following three lemmas can be found, for example, in [6, 14, 30]. Lemma 2.5 Let G be a finite simple group such that G is (lX, mY, nZ)-generated. Then G is (lX, lX, · · · , lX , (nZ)m )-generated, where (nZ)m = {g m | g ∈ nZ}.    m−times

Lemma 2.6 Let G be a finite simple (2X, mY, nZ)-generated group. Then G is (mY, mY, (nZ)2 )-generated. The following results are in some cases useful in establishing non-generation for finite groups. Lemma 2.7 Let G be a finite centerless group. If Δ∗G (C1 , · · · , Ck ) < |CG (gk )|, gk ∈ Ck , then Δ∗G (C1 , C2 , · · · , Ck ) = 0 and therefore G is not (C1 , C2 , · · · , Ck )generated.

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Theorem 2.8 (Ree [70]) Let G be a transitive permutation group generated by permutations g1 , g2 , · · · , gs acting on a set of n elements such  that g1 g2 · · · gs = 1G . If the generator gi has exactly ci cycles for 1 ≤ i ≤ s, then si=1 ci ≤ (s − 2)n + 2. Theorem 2.9 (Scott’s Theorem [30] and [71]) Let g1 , · · · , gs be elements generating a group G with g1 g2 · · · gs = 1G and V be an irreducible module for G with dim V = n ≥ 2. Let CV (gi ) denote thefixed point space of gi on V and let di be the codimension of CV (gi ) in V. Then si=1 di ≥ 2n. With χ being the ordinary irreducible character afforded by the irreducible module V and 1gi being the trivial character of the cyclic group gi , the codimension di of CV (gi ) in V can be computed using the following formula ([44]):   di = dim(V) − dim(CV (gi )) = dim(V) − χ↓G gi , 1gi = χ(1G ) −

o(gi )−1  1 χ(gij ). | gi |

(4)

j=0

3

Special kinds of generation of finite simple groups

In this section we focus on three generational problems for the finite simple groups that are of interest to the authors. This involves looking at the ranks of conjugacy classes of elements, (p, q, r)-generation and the spread of finite non-abelian simple groups. We will also consider briefly the nX-complementary generation. 3.1

The ranks of conjugacy classes of finite simple groups

Note that if X is a non-identity conjugacy class of elements of a finite simple group G, then G = X . In this subsection we are interested in generating a finite simple group by the minimal number of elements from a given non-trivial conjugacy class of the group. Formally: Definition 3.1 Let G be a finite simple group and X be a non-trivial conjugacy class of G. The rank of X in G, denoted by rank(G:X) is defined to be the minimal number of elements of X generating G. We recall from Zisser [82] that for a finite simple group G, the covering number of G is the smallest integer n such that C n = G, for all non-trivial conjugacy classes C of G and by C n the author meant {c1 c2 · · · cn | c1 , c2 , · · · , cn ∈ C}. One of the applications of ranks of conjugacy classes of a finite group is that they are involved in the computations of the covering number of the finite simple group. The following result is in some cases useful in the computations of ranks of classes of a finite non-abelian simple group. Proposition 3.2 Let G be a finite non-abelian simple group such that G is (lX, mY, nZ)-generated. Then rank(G:lX) ≤ m. Also if G is a (2X, mY, nZ)generated group, then rank(G:mY ) = 2.

Basheer, Moori: On the generation of finite simple groups Proof The two assertions are direct corollaries of Lemmas 2.5 and 2.6.

111 

Remark 3.3 We recall from Remark 2.4 that if ΘG > 0 then certainly G is (C1 , · · · , Ck )-generated. In the case C1 = C2 = · · · = Ck−1 = C then G can be generated by k − 1 elements suitably chosen from C and hence rank(G:C) ≤ k − 1. In [69, 67, 65], the second author showed that rank(F i22 :2A) ∈ {5, 6} and that rank(F i22 :2B) = rank(F i22 :2C) = 3. The work of Hall and Soicher [56] implies that rank(F i22 :2A) = 6. Then in a considerable number of publications (for example but not limited to, see [2, 6, 1, 5, 3, 4, 69]) the second author, Ali and Ibrahim explored the ranks for various sporadic simple groups. In fact the determination of the ranks of the sporadic simple groups is almost completed. With G being a sporadic simple group and nX a non-identity class of G (as listed in the Atlas [31]) the results on sporadic simple groups can be summarized as follows: 1. If G = M and nX is not an involutary class, then rank(G:nX) = 2 unless: (G, nX) ∈ {(J2 , 3A), (HS, 4A), (M cL, 3A), (Ly, 3A), (Co1 , 3A), (F i22 , 3A), (F i22 , 3B), (F i23 , 3A), 



(F i23 , 3B), (F i24 , 3A), (F i24 , 3B), (Suz, 3A)}, where in these cases rank(G:nX) = 3. 2. If G = M and nX is an involutary class, then rank(G:2X) = 3 unless: • (G, 2X) ∈ {(J2 , 2A), (Co2 , 2A), (B, 2A)}, where in these cases we have rank(G:2A) = 4. • (G, 2X) ∈ {(F i22 , 2A), (F i23 , 2A)}, where here we have rank(F i22 :2A) = 6 and rank(F i23 :2A) ∈ {5, 6}. 3. If G = M and nX is not an involutary class, then rank(M:nX) ∈ {2, 3}. 4. If G = M and nX is an involutary class, then rank(M:2X) ∈ {3, 4}. Therefore we can see that only very few cases are remaining as far as sporadic simple groups are concerned. The authors are currently considering some of these remaining cases together with other non-abelian simple groups. We conclude this subsection by giving some general results on the ranks for certain conjugacy classes of elements of the simple alternating group An . These results are due to the authors in [15]. The alternating group An , n ≥ 5 has  n3 conjugacy classes of elements of order 3. The cycle structures of these classes are 3m 1n−3m , 1 ≤ m ≤  n3 . In keeping with Atlas notation defined in [31] elements in class 3A are 3-cycles. In this section we determine the rank of this class in An . We will use (An )[k1 ,k2 ,··· ,kr ] to denote the subgroup of An fixing the points k1 , k2 , · · · , kr and if it fixes a single point ki , we use (An )ki . Lemma 3.4 rank(A5 :3A) = 2. Proof It is easy to see that A5 = (1, 2, 3), (1, 4, 5) and clearly a single 3-cycle cannot generate a transitive subgroup, hence rank(A5 :3A) = 2. 

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Lemma 3.5 rank(An :3A) = 2, for all n ≥ 6. Proof Clearly a pair of 3-cycles cannot generate a transitive subgroup, hence rank(An :3A) = 2 for n ≥ 6.  Lemma 3.6 rank(A6 :3A) = 3. Proof It is easy to see that A6 = (1, 2, 3), (1, 4, 5), (1, 5, 6) and by Lemma 3.5 rank(An :3A) = 2, hence rank(A6 :3A) = 3.  The following result is well-known and is useful in the determination of the ranks of many classes of An . Lemma 3.7 (Jordan’s Symmetric Group Theorem) If a primitive permutation group G is a subgroup of Sn and contains a p-cycle for some prime number p < n − 2, then G ∈ {Sn , An }. 

Proof See for example Wielandt [76].

Now we state and prove an important theorem on the rank of the class 3A of An , n ≥ 5. Theorem 3.8 For the alternating group An , n ≥ 5 we have

n−1 if n is odd, 2 rank(An :3A) = n if n is even. 2 Proof No smaller sets of 3A elements will generate a transitive subgroup, so it is sufficient to prove that

(1, 2, 3), (1, 4, 5), (1, 6, 7), · · · , (1, n − 3, n − 2), (1, n − 1, n) if n is odd, An = (1, 2, 3), (1, 4, 5), (1, 6, 7), · · · , (1, n − 2, n − 1), (1, n − 1, n) if n is even. (5) By considering the subgroup   (1, 2, 3), (1, 2, 3)(1,4,5) , (1, 2, 3)(1,5,4) , (1, 2, 3)(1,6,7) , · · · it is easy to see that the group these permutations generate is 2-transitive and therefore primitive. As a primitive group containing three cycles, it follows from Lemma 3.7 that the above permutations generate the whole of An .  Next we turn to the generation of An by the classes of n-cycles for n odd and (n − 1)-cycles for n even. Now we give an important result showing the ranks for the classes of n-cycles and (n − 1)-cycles of An . Theorem 3.9 For n ≥ 5, we have rank(An :nX) = 2 = rank(An :(n − 1)X), X ∈       {A, B}.

n is odd

n is even

Basheer, Moori: On the generation of finite simple groups

113

Proof First, suppose n is odd. The case n = 5 is easily dealt with separately, so assume n ≥ 7. Let α := (1, 2, · · · , n) and let β := (α−1 (3, 2, 1))(5,7) . By direct calculation we have that β is an n-cycle conjugate to α and that αβ = (1, 3, 2)(4, 6)(5, 7) hence (αβ)2 = (1, 2, 3). By considering the subgroup   2 (1, 2, 3), (1, 2, 3)α , (1, 2, 3)α , · · · it is easy to see that the group these permutations generate is 2-transitive and therefore primitive. As a primitive group containing three cycles, it follows from Jordan’s Symmetric Group Theorem that the above permutations generate the whole of An . The argument for n even is similar.  The structure constant method together with the above results and the results of Section 2 were used in [15] to determine the ranks of the classes of alternating groups A8 and A9 . Tables listing the values of h(g, H) (defined in Theorem 2.2) for all the maximal subgroups H of A8 and A9 and all non-trivial classes of these groups, were given in [15]. We also computed all the values of the codimensions di (defined by Equation 4) with respect to the complex irreducible modules of dimensions 7 and 14 of A8 and the complex irreducible module of dimension 8 of A9 . Using Atlas notation as defined in [31], we concluded that for A8 , we have rank(A8 :2A) = rank(A8 :2B) = rank(A8 :3A) = 4 and rank(A8 :nX) = 2, for all nX ∈ {1A, 2A, 2B, 3A}, while for A9 , we have rank(A9 :nA) = 4 for n ∈ {2, 3}, rank(A9 :2B) = 3 and rank(A9 :nX) = 2 for all nX ∈ {1A, 2A, 2B, 3A}. Remark 3.10 Similar to the proof of Theorem 3.9, we can calculate the rank of any p-cycle class in An , for any odd prime p. For example the rank of class 5A (5-cycles) of An for all n ≥ 6 is given by  n! if n = 4k or 4k + 1, 4 (6) rank(An :5A) = "n# if n = 4k + 2 or 4k + 3. 4 Moreover ⎧ (1, 2, 3, 4, 5), (1, 5, 6, 7, 8), (1, 9, 10, 11, 12), · · · , ⎪ ⎪ ⎪ ⎪ (1, n − 6, n − 5, n − 4, n − 3), (1, n − 3, n − 2, n − 1, n) ⎪ ⎪ ⎪ ⎪ (1, 2, 3, 4, 5), (1, 6, 7, 8, 9), (1, 10, 11, 12, 13), · · · , ⎪ ⎪ ⎨ (1, n − 7, n − 6, n − 5, n − 4), (1, n − 3, n − 2, n − 1, n) An = ⎪ (1, 2, 3, 4, 5), (1, 3, 4, 5, 6), (1, 7, 8, 9, 10), · · · , ⎪ ⎪ ⎪ ⎪ (1, n − 4, n − 3, n − 2, n − 1), (1, n − 3, n − 2, n − 1, n) ⎪ ⎪ ⎪ ⎪ (1, 2, 3, 4, 5), (1, 4, 5, 6, 7), (1, 8, 9, 10, 11), · · · , ⎪ ⎩ (1, n − 5, n − 4, n − 3, n − 2), (1, n − 3, n − 2, n − 1, n)

3.2

if n = 4k, if n = 4k + 1, if n = 4k + 2, if n = 4k + 3. (7)

The (p, q, r)- and nX-complementary generation of finite simple groups

We recall that for l, m, n ∈ N \ {1}, a group G is said to be (l, m, n)-generated if G can be generated by two elements x and y with o(x) = l, o(y) = m and

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o(xy) = n. In this case G is a quotient group of the triangle group Δ(l, m, n), where for k1 , k2 , · · · , kr ∈ N \ {1}, the group Δ(k1 , k2 , · · · , kr ) has the presentation   Δ(k1 , k2 , · · · , kr ) = x1 , x2 , · · · , xr | xk11 = xk22 = · · · = xkr r = x1 x2 · · · xr = 1 . If G is (l, m, n)-generated group, then it is also (l , m , n )-generated for any permutation (l , m , n ) of the triple (l, m, n). Thus we may assume that l ≤ m ≤ n. It is well-known that Δ(l, m, n) is finite if and only if 1/l + 1/m + 1/n > 1 (see [32] and [64]). The finite Δ(l, m, n) are: • Δ(1, n, n) the cyclic group of order n, • Δ(2, 2, n) ∼ = D2n the dihedral group of order 2n, • Δ(2, 3, 3) the alternating group A4 , • Δ(2, 3, 4) the symmetric group S4 and • Δ(2, 3, 5) the alternating group A5 . If 1/l + 1/m+ 1/n = 1, namely in the cases of Δ(2, 3, 6), Δ(2, 4, 4) and Δ(3, 3, 3), then the triangle group is infinite but soluble. In the case when 1/l+1/m+1/n < 1, then the triangle group is infinite and insoluble. These triangle groups have a remarkable wealth of interesting finite quotient groups. One can consult [32], [72] and [74] for discussion and background material on triangle groups. The second author and his research group were interested in the (l, m, n)-generation when l, m and n are primes and we adopt the notation (p, q, r)-generation of finite simple groups. We remark that a (2, 3, 7)-generated group G gives rise to compact Riemann surfaces of genus greater than 2 with automorphism groups of maximal order. The (2, 3, 7)-generated groups are called Hurwitz groups ([58] and [74]). The papers by Conder [29, 28] give updates on finite simple groups that are Hurwitz. Related to the (l, m, n)-generation of a group G is the nX-complementary generation of G. Let nX denote a general conjugacy class of G containing elements of order n. A group G is said to be nX-complementary generated if, given an arbitrary non-identity element x ∈ G, there exists a y ∈ nX such that G is x, y . Woldar [80] proved that every sporadic simple group G is pA-complementary generated, where p is the largest prime divisor of |G|. In a series of papers, [49, 48, 46, 47, 45, 68, 66], the second author and Ganief established all possible (p, q, r)-generation and nX-complementary generation, where p, q and r are distinct primes, of the sporadic groups J1 , J2 , J3 , J4 , HS, M cL, Co3 , Co2 and F i22 . Also, the (p, q, r)-generation and nX-complementary gener ation were computed for the sporadic groups Co1 , T h, O N, Ly, Suz and He in [11, 12, 9, 10, 8, 13, 34, 38, 39, 40, 36, 37, 35, 33, 63]. In [16] the authors with Seretlo also established all the (p, q, r)-generation of the Matheiu group M22 . The structure constant method has been used extensively by all the above authors in establishing the (p, q, r)-generation and nX-complementary generation of finite non-abelian simple groups.

Basheer, Moori: On the generation of finite simple groups 3.3

115

The exact spread of finite simple groups

It was shown by Binder in [18, 17] that for any two non-trivial elements x1 and x2 of the symmetric group Sn , n > 4, there exists a third element y such that Sn = x1 , y = x2 , y . This work inspired Brenner and Wiegold [22] to define the spread of a group G. Let r be any positive integer. A finite non-abelian group G is said to have spread r, if for every set {x1 , x2 , · · · , xr } of distinct non-trivial r elements of G, there exists an element y in G such that xi , y = G for all i. It is evident that if G has spread r, then it also has spread k for all k < r, but not conversely. We say that G has exact spread r, denoted by s(G), if it has spread r but not r + 1. The concepts of spread and exact spread are of interest and have many applications to computational group theory (see for example [52]) and also when studying the generating graph of a group (see [61]). Following Brenner and Wiegold [22], the exact spread of the alternating groups is known for even degrees, while for the odd degrees the problem is still open and in this case s(A2n+1 ) tends to infinity as n grows. For groups of Lie type, the exact spread is known in only a few cases such as P SL(2, F). Finally the exact spread of the sporadic groups is known in only two cases, namely the Mathieu Group M11 (see [20, 81]) and in a paper [42] by B. Fairbairn for the Mathieu Group M23 . From [81, 42], we have s(M11 ) = 3 while s(M23 ) = 8064. There have been several improvement to the lower and upper bound of the exact spread by many different authors including the second author (see for example [20, 21, 23, 43, 50, 54]). Most of these authors used probabilistic methods to establish reasonable lower and upper bounds for s(G) for each of the sporadic simple groups. In fact the authors of [23] classified the simple groups G such that s(G) = 2. A related notion is that of having uniform spread which is the same as the concept of spread but with some restrictions. Further information about this concept can be found in [26, 57].

Acknowledgment The authors would like to thank the referee for his/her very valuable corrections, comments and suggestions which contributed significantly in improving the paper. The first author would like to thank the North-West University and the National Research Foundation (NRF) of South Africa for the financial support received. References [1] F. Ali, On the ranks of O’N and Ly, Discrete Appl. Math. 155 (2007), no. 3, 394–399. [2] F. Ali, On the ranks of F i22 , Quaest. Math. 37 (2014), 591–600. [3] F. Ali and M. A. F. Ibrahim, On the ranks of Conway group Co1 , Proc. Japan Acad. Ser. A Math. Sci. 81 (2005), no. 6, 95–98. [4] F. Ali and M. A. F. Ibrahim, On the ranks of Conway groups Co2 and Co3 , J. Algebra Appl. 4 (2005), no. 5, 557–565. [5] F. Ali and M. A. F. Ibrahim, On the ranks of HS and McL, Util. Math. 70 (2006), 187–195. [6] F. Ali and J. Moori, On the ranks of Janko groups J1 , J2 , J3 and J4 , Quaest. Math. 31 (2008), 37–44.

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[7] M. Aschbacher and R. Guralnick, Some applications of the first cohomology group, J. Algebra 90 (1984), 446–460. [8] A. R. Ashrafi, Generating pairs for the Held group He, J. Appl. Math. Comput. 10 (2002), 167–174. [9] A. R. Ashrafi, (p, q, r)-generation and nX-complementary generation of the Thompson group Th, SUT J. Math. (39) (2003), 41–54. [10] A. R. Ashrafi, nX-complementary generation of the Harada-Norton group HN, Acta Math. Inform. Univ. Ostraviensis 11 (2003), 3–9. [11] A. R. Ashrafi, (p, q, r)-generation of sporadic group HN, Taiwanese J. Math. 10 (2006), no. 3, 613–629. [12] A. R. Ashrafi and A. Iranmanesh, nX-complementary generation of the sporadic group Ru, Vietnam J. Math. 33 (2005), no. 1, 1–7. [13] A. R. Ashrafi and G. A. Moghani, nX-complementary generation of the Suzuki group Suz, Bul. Acad. S ¸ tiint¸e Repub. Mold. Mat. 40 (2002), 61–70. [14] A. B. M. Basheer and J. Moori, On the ranks of finite simple groups, Khayyam J. Math. 2 (2016), no. 1, 18–24. [15] A. B. M. Basheer and J. Moori, On the ranks of the alternating group An , Bull. Malays. Math. Sci. Soc., to appear. DOI 10.1007/s40840-017-0586-5. [16] A. B. M. Basheer, J. Moori and T. T. Seretlo, The (p, q, r)-generation of the Matheiu group M22 , Southeast Asian Bull. Math., to appear. [17] G. Ja. Binder, The bases of the symmetric group, Izv. Vysˇs. Uˇcebn. Zaved. Matematika (78), no. 11 (1968), 19–25. [18] G. Ja. Binder, The two-element bases of the symmetric group, Izv. Vysˇs. Uˇcebn. Zaved. Matematika (92), no. 1 (1970), 9–11. [19] W. Bosma and J. J. Cannon, Handbook of Magma Functions (Department of Mathematics, University of Sydeny, November 1994). [20] J. D. Bradley and P. E. Holmes, Improved bounds for the spread of sporadic groups, LMS J. Comput. Math. 10, (2007), 132–140. [21] J. D. Bradley and J. Moori, On the exact spread of sporadic simple groups, Comm. Algebra 35 (2007), no. 8, 2588–2599. [22] J. L. Brenner and J. Wiegold, Two-generator groups. I, Michigan Math. J. 22 (1975), 53–64. [23] T. Breuer, R. Guralnick and W. Kantor, Probabilistic generation of finite simple groups. II, J. Algebra 320 (2008), 443–494. [24] T. Brooks, Generating Sets of Mathieu Groups (PhD Thesis, Cornell University, 2013). [25] T. C. Burness, M. W. Liebeck and A. Shalev, Generation and random generation: from simple groups to maximal subgroups, Adv. Math. 248 (2013), 59–95. [26] T. C. Burness and S. Guest, On the uniform spread of almost simple linear groups, Nagoya Math. J. 209 (2013), 35–109. [27] P. J. Cameron and P. Cara, Independent generating sets and geometries for symmetric groups, J. Algebra 258 (2002), 641–650. [28] M. D. E. Conder, Hurwitz groups: a brief survey, Bull. Amer. Math. Soc. 23 (1990), 359–370. [29] M. D. E. Conder, An update on Hurwitz groups, Groups Complex. Cryptol. 2 (2010), no. 1, 35–49. [30] M. D. E. Conder, R. A. Wilson and A. J. Woldar, The symmetric genus of sporadic groups, Proc. Amer. Math. Soc. 116 (1992), 653–663. [31] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups (OUP, Oxford 1985). [32] H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups (Springer-Verlag, 1980).

Basheer, Moori: On the generation of finite simple groups

117

[33] M. R. Darafsheh and A. R. Ashrafi, (2, p, q)-generation of the Conway group Co1 , Kumamoto J. Math. 13 (2000), 1–20. [34] M. R. Darafsheh and A. R. Ashrafi, Generating pairs for the sporadic group Ru, J. Appl. Math. Comput. 12 (2003), 143–154. [35] M. R. Darafsheh, A. R. Ashrafi and G. A. Moghani, (p, q, r)-generation of the Conway group Co1 , for odd p, Kumamoto J. Math. 14 (2001), 1–20. [36] M. R. Darafsheh, A. R. Ashrafi and G. A. Moghani, (p, q, r)-generation and nX-complementary generation of the sporadic group Ly, Kumamoto J. Math. 16 (2003), 13–25. [37] M. R. Darafsheh, A. Ashrafi and G. Moghani, (p, q, r)-generation of the sporadic group O’N, Groups St Andrews 2001 in Oxford, Vol. I, 101–109, London Math. Soc. Lecture Note Ser. 304 (CUP, Cambridge, 2003). [38] M. R. Darafsheh, A. R. Ashrafi and G. A. Moghani, nX-Complementary generation of the Conway group Co1 , Acta Math. Vietnam. 29 (2004), no. 1, 57–75. [39] M. R. Darafsheh, A. R. Ashrafi and G. A. Moghani, (p, q, r)-generation of the sporadic group O’N, Southeast Asian Bull. Math. 28 (2004), 1–8. [40] M. R. Darafsheh and G. A. Moghani, nX-complementary generation of the sporadic group He, Ital. J. Pure Appl. Math. 16 (2004), 145–154. [41] L. Di Martino, M. Pellegrini and A. Zalesski, On generators and representations of the sporadic simple groups, Comm. Algebra 42 (2014), 880–908. [42] B. Fairbairn, The exact spread of M23 is 8064, Int. J. Group Theory 1 (2012), no. 1, 1–2. [43] B. Fairbairn, New upper bounds on the spreads of the sporadic groups, Comm. Algebra 40 (2012), no. 5, 1872–1877. [44] S. Ganief, 2-Generation of the sporadic simple groups, (PhD Thesis, University of Natal, South Africa, 1997). [45] S. Ganief and J. Moori, rX-complementary generation for the Janko groups J1 , J2 and J3 , Comm. Algebra 24 (1996), 809–820. [46] S. Ganief and J. Moori, (p, q, r)-generation of the smallest Conway group Co3 , J. Algebra 188 (1997), 516–530. [47] S. Ganief and J. Moori, (p, q, r)-generation and nX-complementary generation of the sporadic groups HS and M cL, J. Algebra 188 (1997), 531–546. [48] S. Ganief and J. Moori, Generating pairs for the Conway groups Co2 and Co3 , J. Group Theory 1 (1998), 237–256. [49] S. Ganief and J. Moori, 2-Generation of the forth Janko group J4 , J. Algebra 212 (1999), 305–322. [50] S. Ganief and J. Moori, On the spread of the sporadic simple groups, Comm. Algebra 29 (2001), no. 8, 3239–3255. [51] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.4.10 ; 2007. (http://www.gap-system.org) [52] S. Garion, Connectivity of the product replacement graph of PSL(2,q), J. Group Theory 11 (2008), no. 6, 765–777. [53] R. Guralnick, The spread of a finite simple group, preprint. [54] R. Guralnick and W. Kantor, Probabilistic generation of finite simple groups, J. Algebra 234 (2000), 743–792. [55] R. Guralnick, M. Liebeck, J. Saxl and A. Shalev, Random generation of finite simple groups, J. Algebra 219 (1999), 345–355. [56] J. I. Hall and L. H. Soicher, Presentations of some 3-transposition groups, Comm. Algebra 23 (1995), 2517–2559. [57] S. Harper, On the uniform spread of almost simple symplectic and orthogonal groups, J. Algebra, (2017), in press. [58] A. Hurwitz, Uber algebraische Gebilde mit eindeutigen Transformationen in sich,

118

Basheer, Moori: On the generation of finite simple groups

Math. Ann. 41 (1893), 408–442. [59] P. J. Keen Independent Sets in Some Classical Groups of Dimension Three (PhD Thesis, University of Birmingham, 2011). [60] C. S. King, Generation of finite simple groups by an involution and an element of prime order, J. Algebra 478 (2017), 153–173. [61] A. Lucchini and A. Maroti, Some results and questions related to the generating graph of a finite group, Ischia group theory 2008, 183–208, World Sci. Publ., Hackensack, NJ, 2009. [62] K. Lux and H. Pahlings, Representations of Groups: A Computational Approach (CUP, Cambridge, 2010). [63] K. Mehrabadi, A. R. Ashrafi and A. Iranmanesh, (p, q, r)-Generation of the Suzuki group Suz, Int. J. Pure Appl. Math. 11 (2004), no. 4, 447–463. [64] G. A. Miller, Groups defined by the orders of two generators and the order of their product, Amer. J. Math. 24 (1902), 96–100. [65] J. Moori, Generating sets for F22 and its automorphism group, J. Algebra 159 (1993), 488–499. [66] J. Moori, (p, q, r)-generation for the Janko groups J1 and J2 , Nova J. Algebra Geom. 2 (1993), no. 3, 277–285. [67] J. Moori, Subgroups of 3-transposition groups generated by four 3-transpositions, Quaest. Math. 17 (1994), 483–494. [68] J. Moori, (2, 3, p)-generation for the Fischer group F22 , Comm. Algebra 22 (1994), no. 11, 4597–4610. [69] J. Moori, On the ranks of the Fischer group F22 , Math. Japon. 43 (1996), 365–367. [70] R. Ree, A theorem on permutations, J. Combin. Theory Ser. A 10 (1971), 174–175. [71] L. L. Scott, Matrices and cohomology, Ann. Math. 105(3) (1977), 473–492. [72] D. Singerman, Subgroups of Fuchsian groups and finite permutation groups, Bull. London Math. Soc. 2 (1970), 319–323. [73] R. Steinberg, Generators for simple groups, Canad. J. Math. 14 (1962), 277–283. [74] T. W. Tucker, Finite groups acting on surfaces and the genus of a group, J. Comb. Theory B 34 (1983), 82–98. [75] J. Ward, Generation of Simple Groups by Conjugate Involutions (PhD Thesis, University of London, 2009). [76] H. Wielandt, Finite Permutation Groups (New York: Academic Press, 1964). [77] J. Whiston, Maximal independent generating sets of the symmetric group, J. Algebra 232 (2000), no. 1, 255–268. [78] J. Whiston, The Minimal Generating Sets of Maximal Size of Selected Groups (PhD thesis, University of Cambridge, 2001). [79] J. Whiston and J. Saxl, On the maximal size of independent generating sets of P SL2 (q), J. Algebra, 258 (2002), 651–657. [80] A. Woldar, 3/2-generarion of the sporadic simple groups, Comm. Algebra 22 (1994), no. 2, 675–685. [81] A. Woldar, The exact spread of the Mathieu group M11 , J. Group Theory 10 (2007), 167–171. [82] I. Zisser, The covering numbers of the sporadic simple groups, Israel J. Math. 67 (1989), 217–224.

ONE-RELATOR GROUPS: AN OVERVIEW GILBERT BAUMSLAG∗ , BENJAMIN FINE† and GERHARD ROSENBERGER§ ∗

Department of Computer Science, City College of New York, New York, NY 10031 Department of Mathematics, Fairfield University, Fairfield, Connecticut 06430 § Fachbereich Mathematik, University of Hamburg, Hamburg 20146, Germany †

In memory of Gilbert Baumslag Sadly, our coauthor and friend Gilbert Baumslag died during the preparation of this paper. The remaining two authors dedicate the paper to the memory of Gilbert and to all the work that he inspired.

Abstract In 1985, at Groups St Andrews, Gilbert Baumslag gave a short course on onerelator groups which provided a look at the subject up to that point. In this paper we partially update the massive amount of work done over the past three decades. For the most part we concentrate on areas and results to which the authors have made contributions. We look at the important connections with surface groups and elementary theory, and describe the surface group conjecture and the Gromov conjecture on surface subgroups. We look at the solution by Wise of Baumslag’s residual finiteness conjecture and discuss a new Baumslag conjecture on virtually free-by-cyclic groups. We examine various amalgam decompositions of one-relator groups and the Baumslag-Shalen conjectures. We then look at a series of open problems in one-relator group theory and their status. Finally we introduce a concept called plainarity based on the Magnus breakdown of a one-relator group which might provide a systematic approach to the solution of problems in onerelator groups.

1

Introduction

One-relator groups have always played a fundamental role in combinatorial group theory. This is true for a variety of reasons. From the viewpoint of presentations they are the simplest groups after free groups which they tend to resemble in structure. Secondly as a class of groups they have proved to be somewhat amenable to study. However most important is that they arise naturally in the study of low-dimensional topology, specifically as fundamental groups of two-dimensional surfaces. As explained in [2] surface groups have served as motivating examples for much of combinatorial group theory. At Groups St Andrews 1985 Gilbert Baumslag gave a series of lectures on onerelator groups and the survey article [10] provided an in-depth view of the diversity of topics that constituted one-relator group theory up to 1985. However over the past three decades there have been tremendous strides in combinatorial group

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theory in general, and one-relator group theory in particular, stimulated by ideas from hyperbolic geometry, algebraic geometry and logic. Many of these ideas have coalesced into a new part of infinite group theory called geometric group theory. The concept of a hyperbolic group, now central to the field, can be traced back to the work of Max Dehn on fundamental groups of compact two-dimensional surfaces, which are all one-relator groups. These ideas from hyperbolic geometry have also proved important in pointing out links between finitely presented groups and recursive function theory. They also provide a starting point for a the study of certain groups with two defining relations – witness the work of Rapaport on the Poincar´e conjecture [118]. Indeed concentration on the residual freeness [6] of certain one-relator groups provides an entr´ee into such a study. The development of algebraic geometry over groups by Baumslag, Myasnikov and Remeslennikov [18, 19] also has close ties to one-relator group theory. Algebraic geometry over groups has led to the solution of the celebrated Tarski conjectures by Kharlampovich and Myasnikov [88,89,91–94] and independently by Sela [127–132]. Here surface groups serve as the primary examples of non-free elementary free groups, that is non-free groups that have the same elementary theory as free groups. We will say more about this in Section 2. In addition to the study of one-relator groups themselves, there has been a large body of research over the past three decades on one-relator products and generalizations of the classical Freiheitssatz. We refer to the article [63] and the book [64] for a comprehensive discussion of these. This paper is a partial updating of Baumslag’s classic survey. We first look at the present state of one-relator group theory in light of geometric group theory and algebraic geometry over groups. We then examine a collection of open problems on one-relator groups. Finally we propose a method called plainarity which uses the Magnus breakdown as a proposed systematic method to study groups with one defining relator. The outline of this paper is as follows. In the next section we discuss surface groups and the ties to geometric group theory and elementary group theory and in particular the role they play in the solution of the Tarski problems. From this we talk about elementary free groups in general and the techniques they provide to prove non-trivial results about surface groups. Related to this we discuss the surface group conjecture, Gromov’s surface subgroup conjecture, Baumslag doubles and the classification of fully residually free one-relator groups. In Section 3 we consider Baumslag’s residual finiteness conjecture that a onerelator group with torsion is residually finite. We discuss the brilliant geometric solution given by Wise [143]. We also describe a newer Baumslag conjecture called the virtually free-by-cyclic conjecture. A result of Baumslag and Shalen [21] and improved by Fine, Levin and Rosenberger [55] and independently by Benyash-Krivets [24] in a more general context, shows that any one-relator group with more than two generators or with two generators and torsion has a free product with amalgamation decomposition. In Section 4 we look at this question and discuss several conjectures on what are termed Baumslag-Shalen decompositions.

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In Section 4 we look at a collection of open problems on one-relator groups and talk about the status of each problem. Despite the large interest and increasing importance of one-relator groups, as well as the many new ideas that have resulted from their study, no systematic investigation of these groups has been undertaken. The survey mentioned above, the book by Fine and Rosenberger [64] and the large body of work that already exists, suggest that such an investigation is appropriate at this time. The surprisingly complex structure of one-relator groups that became apparent with the discovery, by Baumslag and Solitar in 1961, that some of them could be isomorphic to their proper factor groups [22], underlines the difficulty of such an endeavour. Despite these and other difficulties, in Section 5 we propose a method to initiate a systematic study of one-relator groups based primarily on an in-depth study of the Magnus breakdown of one-relator groups and a concept that we call plainarity. This concept begins with those one-relator groups that can be constructed in one step from free products of cyclic groups. Plainarity, coupled with the use of a decomposition theorem of Baumslag and Shalen [21], the machinery of groups acting on trees, and intensive use of both the Reidemeister-Schreier procedure and computational methods provides a wide array of tools for the proposed systematic investigation.

2

Surface Groups, Hyperbolic Groups and Elementary Theory

Much of the theory of one-relator groups, as well as much of combinatorial group theory in general, has been motivated by the properties of surface groups. This was written about in detail in [2]. As new ideas such as hyperbolic groups and elementary free groups arose in group theory the important ties to surface groups continued. In this section we discuss some important results on surface groups which are most relevant to these new developments. Recall that an orientable surface group Sg is the fundamental group of an orientable compact surface of genus g. Such a group has a one-relator presentation Sg = a1 , b1 , . . . , ag , bg ; [a1 , b1 ] . . . [ag , bg ] = 1

with g ≥ 1.

A nonorientable surface group Tg is the fundamental group of a nonorientable compact surface of genus g. Such a group also has a one-relator presentation, now of the form Tg = a1 , a2 , . . . , ag ; a21 a22 · · · a2g = 1 with g ≥ 1. Fricke and Klein [69] building on work of Poincar´e proved that in the orientable case the groups Sg for all g and Tg for g ≥ 4 have faithful representations in P SL2 (C). It follows from a theorem of Mal’cev [Ma] that each group is residually finite. Recall that a group G is residually finite if given any element g ∈ G, g = 1, there exists a normal subgroup N of finite index in G such that g ∈ / N . It follows that G has a solvable word problem. Dehn [42] proved that the fundamental group of an orientable compact surface of genus g ≥ 2 has a solvable word problem by showing that if any cyclically reduced word w is equal to 1 in Sg then more than

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half u of one of the cyclic conjugates uv −1 of [a1 , b1 ] . . . [ag , bg ] or its inverse occurs in w. On replacing u by v in w, the resultant shorter word w is also equal to 1 in Sg and so the process can be repeated, ultimately leading to a proof that w is equal to 1 in Sg . This algorithm has been coined Dehn’s algorithm. Hyperbolic groups can be defined as those groups with a finite presentation where Dehn’s algorithm solves the word problem (see for example the books [31] or [36]). Dehn solved the conjugacy problem for Sg in a similar manner; again a similar argument can be used to solve the conjugacy problem for every hyperbolic group (see [75]). The class of hyperbolic groups is contained in a somewhat wider class, the class of automatic groups (see [45]). We recall that a group G is termed automatic if it has a finite set X of semigroup generators and there is regular language L over X such that (i) every element of G is equal in G to some word in L, and (ii) for each generator x ∈ X there is a finite automaton verifying right multiplication of the elements of L by that generator. In many cases the regular language consists of the set of words that are minimal with respect to the ShortLex ordering. 2.1

Cyclically Pinched and Conjugacy Pinched One-relator Groups

If g ≥ 2 then each surface group Sg has a free product with amalgamation decomposition of the form Sg = F1  F2 U =V

where F1 is the free group on a1 , b1 , . . . , ag−1 , bg−1 , F2 is the free group on ag , bg and U = [a1 , b1 ] . . . [ag−1 , bg−1 ], V = ([ag , bg ])−1 . In general a cyclically pinched one-relator group is a group with a finite presentation of the form G = F1  F2 U =V

where F1 , F2 are free groups and U, V represent nontrivial elements in the respective free groups. Hence any orientable surface group of genus g ≥ 2 falls in the larger class of cyclically pinched one-relator groups. A conjugacy pinched one-relator group is the HNN analog of a cyclically pinched one-relator group. This is a group with a finite presentation of the form G = t, F ; t−1 U t = V where F is a free group and U, V are nontrivial elements in F . An orientable surface group with g ≥ 2 can also be expressed as a conjugacy pinched one-relator group (see the book [48]). Before continuing we mention when groups of these types are actually free groups. For cyclically pinched groups this follows from the primitivity of elements generating the amalgamated subgroups.

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Theorem 2.1 Let G be a cyclically pinched one-relator group so that G = a1 , . . . , ap , b1 , . . . , bq ; U = V where U ∈ a1 , . . . , ap , V ∈ b1 , . . . , bq . Then G is a free group if and only if U is primitive in a1 , . . . , ap or V is primitive in b1 , . . . , bq . The situation is a bit more complicated for conjugacy pinched one-relator groups. Theorem 2.2 Let G be a conjugacy pinched one-relator group so that G = a1 , . . . , ap , t ; tU t−1 = V where U, V ∈ a1 , . . . , ap with p ≥ 1 and U = 1, V = 1. Then G is a free group if and only if one of the following holds: (1) a1 , . . . , ap has a basis {U, x1 , . . . , xp−1 } a1 , . . . , ap to some V1 ∈ x1 , . . . , xp−1

such that

V is conjugate in

(2) a1 , . . . , ap has a basis {V, x1 , . . . , xp−1 } a1 , . . . , ap to some U1 ∈ x1 , . . . , xp−1

such that

U is conjugate in

Cyclically pinched and conjugacy pinched one-relator groups share many general properties with surface groups. This is especially true with linearity results, that is results also shared by linear groups. Wehfritz [138] showed that a cyclically pinched one-relator group where neither U nor V is a proper power has a faithful representation over a commutative field and is hence linear. Using a result of Shalen [134] and generalized by Fine and Rosenberger [61] if neither U nor V is a proper power then a cyclically pinched one-relator group has a faithful representation in P SL2 (C) (see [66] and [67]). Further under the same conditions Fine, Kreuzer and Rosenberger [54] showed that there is faithful representation in P SL2 (R). We will say more about this in Section 4. In particular cyclically pinched one-relator groups are residually finite and coherent, that is finitely generated subgroups are finitely presented, a result originally due to Karrass and Solitar [87]. We summarize many of these results in the following theorem. Theorem 2.3 Let G be a cyclically pinched one-relator group. Then (1) G is residually finite ([10]). (2) G has a solvable conjugacy problem ([101]) and is conjugacy separable ([43]). (3) G is subgroup separable ([33]) and generalized in ([1]). (4) If neither U nor V is a proper power then G has a faithful representation over some commutative field ([138]). (5) If neither U nor V is a proper power then G has a faithful representation in P SL2 (C) ([63]) and in P SL2 (R) ([54]). (6) If either U or V is not a proper power then G is hyperbolic ([26, 83, 95]). (7) If either U or V is not a proper power then G has a faithful representation in P SL2 (R) ([56]).

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(8) If neither U nor V is in the commutator subgroup of its respective factor then G is free-by-cyclic ([14]). (9) In p + q > 2 then G is SQ-universal ([125]). In particular G contains a nonabelian free group. Recall that a group G is SQ-universal if every countable group can be embedded as a subgroup of a quotient of G. SQ-universality is one measure of largeness for an infinite group (see [64]). We remark that the conclusion of (9) is not correct if p + q = 2 and G = a, b ; a2 = b2 ; it does not hold if p + q = 2 and G = a, b ; ar = bs with r, s > 1 and r + s > 4. Rosenberger, using Nielsen cancellation, has given a positive solution to the isomorphism problem for cyclically pinched one-relator groups. Theorem 2.4 ([121]) The isomorphism problem for any cyclically pinched onerelator group is solvable; given a cyclically pinched one-relator group G there is an algorithm to decide in finitely many steps whether an arbitrary one-relator group is isomorphic or not to G. Although not directly connected to cyclically pinched one-relator groups we mention that Dahmani and Guiradel [41] have proved that all one-relator groups with torsion have a solvable isomorphism problem. This is an outgrowth of their solution to the isomorphism problem for hyperbolic groups with torsion. Sela [126] had earlier proved the solvability of the isomorphism problem for torsion-free hyperbolic groups. Conjugacy pinched one-relator groups are the HNN analogs of cyclically pinched one-relator groups and are also motivated by the structure of orientable surface groups Sg . In particular suppose Sg = a1 , b1 , . . . , ag , bg ; [a1 , b1 ] . . . [ag , bg ] = 1

with g ≥ 2.

Let bg = t. Then Sg is an HNN group of the form Sg = a1 , b1 , . . . , ag , t ; tU t−1 = V where U = ag and V = [a1 , b1 ] . . . [ag−1 , bg−1 ]ag . We now discuss a generalization of this. Structurally such a group is an HNN extension of the free group F on a1 , . . . , an with cyclic associated subgroups generated by U and V with both U and V nontrivial and is hence the HNN analog of a cyclically pinched one-relator group. Groups of this type arise in many different contexts and share many of the general properties of the cyclically pinched case. However many of the proofs become tremendously more complicated in the conjugacy pinched case as compared to the cyclically pinched case. Furthermore in most cases additional conditions on the associated elements U and V are necessary. To illustrate this we state a result ([59], see [64]) which gives a partial solution to the isomorphism problem for conjugacy pinched one-relator groups.

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Theorem 2.5 Let G = a1 , . . . , ap , t ; tU t−1 = V be a conjugacy pinched onerelator group and suppose that neither U nor V is a proper power in the free group on a1 , . . . , ap . Suppose further that there is no Nielsen transformation from {a1 , . . . , ap } to a system {b1 , . . . , bp } with U ∈ {b1 , . . . , bp−1 } and that there is no Nielsen transformation from {a1 , . . . , ap } to a system {c1 , . . . , cp } with V ∈ {c1 , . . . , cp−1 }. Then: (1) G has rank p + 1 and for any minimal generating system for G there is a one-relator presentation. (2) The isomorphism problem is solvable. More information about both cyclically pinched one-relator groups and conjugacy pinched one-relator groups is in [64] or [67]. A group G is equationally Noetherian if the set of all solutions of any system of equations over G in finitely many variables is the solution set of a finite system of equations over G. Guba [76] proved that any linear group over a commutative field is equationally Noetherian. Hence orientable surface groups Sg and more generally a larger subclass of cyclically pinched and conjugacy pinched one-relator groups are equationally Noetherian. From a result of Mal’cev [110] the residual finiteness of the groups Sg implies that they are hopfian, that is they cannot be isomorphic to a proper factor group. 2.2

Surface Groups and Elementary Theory

The groups Sg and Tg are also heavily involved in the elementary theory of groups. Here we first must recall a collection of definitions both from group theory and logic. Definition 2.6 A group G is residually free if for each non-trivial g ∈ G there is a free group Fg and an epimorphism hg : G → Fg such that hg (g) = 1. Equivalently for each g ∈ G there is a normal subgroup Tg such that G/Tg is free and g ∈ / Tg . The group G is fully residually free provided to every finite set S ⊂ G \ {1} there is a free group FS and an epimorphism hS : G → FS such that hS (g) = 1 for all g ∈ S. A result of G. Baumslag [6] showed that each Sg is residually free. Combining this with a result of B. Baumslag [5] we get that each Sg is fully residually free as well. Theorem 2.7 For all g ≥ 1 the surface group Sg of genus g is fully residually free. In [6] a more general result was proved. If F is a nonabelian free group and u ∈ F is a nontrivial element which is neither primitive nor a proper power then the one-relator group K given by K=F  F u=u

where F is an identical copy of F and u is the corresponding element to u in F , is called a Baumslag double.

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Theorem 2.8 Any Baumslag double is fully residually free. The class of finitely generated fully residually free groups was introduced in a different research direction by Sela in his solution of the Tarski problems. In Sela’s approach these groups appear as limits of homomorphisms of a group G into a free group. In this guise they are called limit groups. Therefore a limit group is a finitely generated fully residually free group (see [26]) and [48] for nice descriptions of the equivalence of the two approaches. Fully residually free groups are tied to logic and the elementary theory of groups in the following manner. We start with a first-order language appropriate for group theory. This language which we denote by L0 is the first-order language with equality containing a binary operation symbol · a unary operation symbol −1 and a constant symbol 1. A sentence in this language is a logical expression containing a string of variables x = (x1 , . . . , xn ), the logical connectives ∨, ∧, ∼ and the quantifiers ∀, ∃. A universal sentence of L0 is one of the form ∀x{φ(x)} where x is a tuple of distinct variables, φ(x) is a formula of L0 containing no quantifiers and containing at most the variables of x. Similarly an existential sentence is one of the form ∃x{φ(x)} where x and φ(x) are as above. If G is a group, then the universal theory of G consists of the set of all universal sentences of L0 which are true in G. We denote the universal theory of a group G by T h∀ (G). Since any universal sentence is equivalent to the negation of an existential sentence it follows that two groups have the same universal theory if and only if they have the same existential theory. The set of all sentences of L0 true in G is called the first-order theory or the elementary theory of G. We denote this by T h(G). We note that being first-order or elementary means that in the intended interpretation of any formula or sentence all of the variables (free or bound) are assumed to take on as values only individual group elements – never, for example, subsets of, nor functions on, the group in which they are interpreted. The Tarski conjectures, solved independently by Kharlampovich and Myasnikov (see [88,89,92–94]) and Sela (see [127–131]), say essentially that all nonabelian free groups have the same elementary theory. The following was well-known and much simpler. Theorem 2.9 All nonabelian free groups have the same universal theory ([48]). As we will see, all finitely generated fully residually free groups will also have the same universal theory as the class of nonabelian free groups. A universally free group G is a group that has the same universal theory as a nonabelian free group. Gaglione and Spellman [71] and independently Remeslennikov [119] proved the following remarkable theorem. Theorem 2.10 Let G be a nonabelian finitely generated group. Then G is fully residually free if and only if G is universally free. Subsequently Myasnikov and Remeslennikov [114] showed that finitely generated fully residually free groups are precisely the finitely generated subgroups of the free exponential group F Z[x] . A group is coherent if all finitely generated subgroups

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are also finitely presented. From the result of Myasnikov and Remeslennikov, Kharlampovich and Myasnikov [95] and independently Sela [126] proved that each fully residually free group is coherent. Free groups and hence fully residually free groups are residually nilpotent and the successive quotients of their lower central series are all free abelian. One can associate to each group G its so-called Lazard Lie ring ([99]). The Lazard Lie ring of a free group is a free Lie ring. The residually nilpotent groups with free Lazard Lie ring are termed parafree ([7]). Parafree groups can be embedded in the group of units of the ring of power series in an appropriate number of non-commuting variables. Since orientable surface groups can be embedded in such parafree groups, they also admit an embedding in the unit groups of power series rings. We discuss parafree groups further in Section 4. 2.3

Elementary Free Groups

An elementary free group is a group having the same elementary theory as the nonabelian free groups. Clearly the class of elementary free groups contains the class of universally free groups and hence the fully residually free groups. The proofs of both Kharlampovich and Myasnikov and Sela completely describe the class of elementary free groups which extends beyond the free groups themselves. The surface groups Sg with genus g ≥ 2 are the primary examples of non-free elementary free groups. Theorem 2.11 The orientable surface groups Sg with g ≥ 2 and the nonorientable surface groups Tg with g ≥ 4 are elementary free. This provides an interesting and powerful technique to prove nontrivial results in surface groups. These have been dubbed something for nothing results. In particular any first-order result on nonabelian free groups is true in any elementary free groups and in particular a surface group. Magnus [108] proved the following often used theorem in free groups. Theorem 2.12 Let F be a nonabelian free group and R, S ∈ F . Then if N (R) = N (S), it follows that R is conjugate to either S or S −1 . Here N (g) denotes the normal closure in F of the element g. Howie [78] and independently Bogopolski [28] and Bogopolski and Sviridov [29] gave a proof of this result for surface groups. Howie’s proof was for orientable surface groups while Bogopolski and Sviridov also handled the nonorientable case. That is Magnus’s theorem holds if the free group F is replaced by a surface group of appropriately high genus. Their proofs were nontrivial and Howie’s proof used the topological properties of surface groups. Howie further developed, as part of his proof of Magnus’s theorem for surface groups, a theory of one-relator surface groups. These are surface groups modulo a single additional relator. Bogopolski and Bogopolski-Sviridov proved in addition that Magnus’s Theorem holds in an even wider class of groups.

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In [49] and [70] it was determined that Magnus’s result is actually a first-order theorem on nonabelian free groups and hence from the theorems concerning the solution of the Tarski problems it holds automatically in all elementary free groups. In particular Magnus’s theorem will hold in surface groups, both orientable and nonorientable of appropriate genus. Magnus’s theorem can be given by a sequence of elementary sentences of the form ([70]) {∀R, S ∈ G, ∀g ∈ G ∃g1 , . . . , gt , h1 , . . . , hk } −1 ±1 ±1 (g −1 Rg = g1−1 S ±1 g1 . . . gt−1 S ±1 gt ) ∧ (g −1 Sg = h−1 1 R h1 . . . hk R hk )

=⇒ {∃x ∈ G(x−1 Rx = S ∨ x−1 Rx = S −1 )}. Magnus’s theorem is therefore a first-order result. Since it is known to be true in nonabelian free groups it will then be true in any elementary free group from the solution to the Tarski problems. Therefore we get the following theorem. Theorem 2.13 Let G be an elementary free group and R, S ∈ G. Then if N (R) = N (S) it follows that R is conjugate to either S or S −1 . It follows that any elementary free group and hence any surface group of the appropriate genus satisfies Magnus’s theorem. This recovers the results of Howie and Bogopolski. Actually more is true. An examination of the sentences capturing the fact that Magnus’s theorem is first-order shows that the sentences are universalexistential. Hence the theorem holds in the almost locally free groups of Gaglione and Spellman [71]. Corollary 2.14 ([28, 78]) Let Sg be an orientable surface group of genus g ≥ 2. Then Sg satisfies Magnus’s theorem, that is if u, v ∈ Sg and N (u) = N (v) it follows that u is conjugate to either v or v −1 . Corollary 2.15 ([29]) Let Tg be a nonorientable surface group of genus g ≥ 4. Then Tg satisfies Magnus’s theorem, that is if u, v ∈ Tg and N (u) = N (v) it follows that u is conjugate to either v or v −1 . The genus g ≥ 4 is essential here. If g < 4 in the nonorientable case the corresponding surface group is not elementary free. Many other nontrivial results on surface groups can be proved in this manner. Further it can be proved that elementary free groups satisfy a collection of properties that are not first-order. These results were done in [49, 50] and in particular hold in the class of surface groups. For example, all elementary free groups are hyperbolic and stably hyperbolic. Further they are all Turner Groups (see [49, 50], that is they have test elements and satisfy Turner’s Retraction Theorem [137]). 2.4

The Surface Group Conjecture

Related to the problem of discerning one-relator groups by the form of their relator is the surface group conjecture. In the Kourovka notebook [112] Melnikov proposed the following problem.

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Question 2.16 Suppose that G is a residually finite non-free, non-cyclic onerelator group such that every subgroup of finite index is again a one-relator group. Must G be a surface group? As asked by Melnikov the answer is no. Recall that the Baumslag-Solitar groups BS(m, n) are the groups BS(m, n) = a, b; a−1 bm a = bn with m = 0 nad n = 0. If |m| = |n| or either |m| = 1 or |n| = 1 these groups are residually finite. They are hopfian if |m| = 1 or |n| = 1 or both m and n have the same prime factors. In all other cases they are non-hopfian. If either |m| = 1 or |n| = 1 then every subgroup of finite index is again a Baumslag-Solitar group and therefore a one-relator group. It follows that besides the surface groups the groups BS(1, m) also satisfy the hypotheses of Melnikov’s question. We then have the following conjecture. Conjecture 2.17 (Surface Group Conjecture A) Suppose that G is a residually finite non-free, non-cyclic one-relator group such that every subgroup of finite index is again a one-relator group. Then G is either a surface group or a BaumslagSolitar group B(1, m) for some integer m. We note that the groups B(1, 1) and B(1, −1) are surface groups. In surface groups, subgroups of infinite index must be free groups and there are noncyclic free groups except if the surface group is isomorphic to B(1, 1) of B(1, −1). This is not true in the groups BS(1, m). To avoid the Baumslag-Solitar groups, Surface Group Conjecture A, was modified to ([38]): Conjecture 2.18 (Surface Group Conjecture B) Suppose that G is a nonfree, non-cyclic one-relator group such that every subgroup of finite index is again a one-relator group and every subgroup of infinite index is a free group and G contains nonabelian free groups as subgroups of infinite index. Then G is a surface group. Using the structure theorem for fully residually free groups in terms of its JSJ decomposition (see [88,89,92–94], [127–131] or [120]), Fine, Kharlampovich, Myasnikov, Remeslennikov and Rosenberger [52] made some progress on these conjectures. Finally Ciobanu, Fine and Rosenberger [38] building on work of H. Wilton [139] settled the surface group conjecture if G is assumed to be either a cyclically pinched one-relator group or a conjugacy pinched one-relator group. We say that a group G satisfies Property IF if every subgroup of infinite index is free. Recall that the standard one-relator presentation for a surface group allows for a decomposition as a cyclically pinched one-relator group and as a conjugacy pinched one-relator group ([67] and [145]). In particular the following results are proved in [52]. Theorem 2.19 Suppose that G is a finitely generated fully residually free group with property IF. Then G is either a free group or a cyclically pinched one relator group or a conjugacy pinched one-relator group.

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Corollary 2.20 Suppose that G is a finitely generated fully residually free group with property IF. Then G is either free or every subgroup of finite index is freely indecomposable and hence a one-relator group. Further if the surface group conjecture is true then a group satisfying the conditions of the conjecture must be hyperbolic or free abelian of rank 2. The following is also proved. Theorem 2.21 ([52]) Let G be a finitely generated fully residually free group with property IF. Then either G is hyperbolic or G is free abelian of rank 2. In light of these results the following modified version of the surface group conjecture was given. Conjecture 2.22 (Surface Group Conjecture C) Suppose that G is a finitely generated non-free freely indecomposable fully residually free group with property IF. Then G is a surface group. Finally we note that although the focus in [52] was on Property IF there has been some evidence for the Surface Group Conjecture based on the subgroups of finite index. Note that an orientable surface group of genus g ≥ 2 with the presentation G = a1 , b1 , . . . , ag , bg ; [a1 , b1 ] . . . [ag , bg ] also has a presentation −1 G = x1 , . . . , xn ; x1 . . . xn x−1 1 . . . xn = 1

with n even. Curran [40] has proved the following. Theorem 2.23 Let G be a one-relator group with the presentation 1 n G = x1 , . . . , xn ; xν11 . . . xνnn x−ν . . . x−ν = 1 . n 1

Then, if n is odd, there exist normal subgroups of finite index which do not have one-relator presentations. In particular if −1 G = x1 , . . . , xn ; x1 . . . xn x−1 1 . . . xn = 1

then every subgroup of finite index is again a one-relator group if and only if n is even and hence G is a surface group. Using the following result of Wilton combined with results of Guildenhuys, Kharlampovich and Myasnikov and the Karrass-Solitar subgroup theorems for free products with amalgamation, Ciobanu, Fine and Rosenberger [38] settled Surface Group Conjecture C and the general conjecture for cyclically pinched and conjugacy pinched one-relator groups.

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Theorem 2.24 ([139]) Let G be a hyperbolic one-ended cyclically pinched onerelator group or a hyperbolic one-ended conjugacy pinched one-relator group. Then either G is a surface group, or G has a finitely generated non-free subgroup of infinite index. Let G be a finitely generated group. Let S ⊂ G be a finite generating set of G and let Γ(G, S) be the Cayley graph of G with respect to S. Then the number of ends of G is defined as e(G) = e(Γ(G, S)). It was proved by Stallings that a finitely generated group G has more than one end if and only if the group G admits a nontrivial decomposition as an amalgamated free product or an HNN extension over a finite subgroup. In [38] it was proved that Surface Group Conjecture C is true. Theorem 2.25 Suppose that G is a finitely generated non-free freely indecomposable fully residually free group with property IF. Then G is a surface group. That is, Surface Group Conjecture C is true. Thus fully residually free and Property IF completely characterize surface groups. Theorem 2.26 A group G is a surface group if and only if G is finitely generated, non-free, indecomposable, fully residually free and satisfies Property IF. The main result in [38] is that the Surface Group Conjecture is true if G is a cyclically pinched or conjugacy pinched one-relator group. Theorem 2.27 (1) Let G be a cyclically pinched one-relator group satisfying property IF. Then G is a free group or a surface group. (2) Let G be a conjugacy pinched one-relator group satisfying property IF. Then G is a free group, a surface group or a solvable Baumslag-Solitar group. We finally remark that Ponzoni [116] had some success with Surface Group Conjecture A for two-generator one-relator groups. 2.5

Gromov’s Surface Subgroup Conjecture

Recall that a Baumslag double is an amalgamated product of the form G=F



{W =W }

F

where F is a finitely generated free group, F is an isomorphic copy of F , W is a word in F and W is its copy in F . The resulting Baumslag double G is hyperbolic if and only if W is not a proper power in F (see Theorem 2.3 (6)). A conjecture of Gromov states that a one-ended word hyperbolic group must contain a subgroup isomorphic to the fundamental group of a closed hyperbolic surface (see [97]). Kim and Oum [97] prove that this is true for any one-ended Baumslag double if (1) the free group has rank 2 or (2) every generator is used the same number of times in a minimal automorphic image of the amalgamating words. This builds on work of

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Gordon and Wilton [74] and Kim and Wilton [98] who gave sufficient conditions for hyperbolic surface groups to be embedded in a Baumslag double G. In order to prove this, Kim and Oum formulate a stronger statement on Whitehead graphs and prove its specialization by combinatorial induction for (1) and the characterization of perfect matching polytopes by Edmonds [44] for (2). In [65] using Nielsen cancellation methods it was proved that a hyperbolic orientable surface group of genus 2 is embedded in a hyperbolic Baumslag double if and only if the amalgamated word W is a commutator, that is W = [U, V ] for some elements U, V ∈ F . Further G contains a nonorientable surface group of genus 4 if and only if W = X 2 Y 2 for some X, Y ∈ F . G can contain no nonorientable surface group of smaller genus, see [65]. If a group contains a hyperbolic surface subgroup of genus 2 it will contain a hyperbolic surface subgroup of any genus g ≥ 2. Using Nielsen cancellation methods Fine and Rosenberger [64] prove the following. Theorem 2.28 Let G = F



{W =W }

F be a hyperbolic Baumslag double. Then

G contains a hyperbolic orientable surface group of genus 2 if and only W is a commutator, that is W = [U, V ] for some elements U, V ∈ F . Further a Baumslag double G contains a nonorientable surface group of genus 4 if and only if W = X 2 Y 2 for some X, Y ∈ F . Since an orientable surface group of genus 2 contains an orientable surface group of any finite genus as a subgroup we immediately get the following corollary. Corollary 2.29 Let G = F



{W =W }

F be a hyperbolic Baumslag double. Then G

contains orientable surface groups of all finite genus if and only W is a commutator. 2.6

The Classification of Fully Residually Free One-Relator Groups

As described earlier, finitely generated fully residually free groups, or limit groups, have played a fundamental role in the study of the elementary theory of groups. This class includes the surface groups and has been extensively studied. In particular a whole theory of infinite words [96] has been developed that mirrors the ordinary theory of words in free groups. From the theory of infinite words the algorithmic solutions to many problems, such as the generalized word problem, have been found for fully residually free groups. There is an approach to studying fully residually free groups somewhat analogous to our proposed plainarity approach. Myasnikov and Remeslennikov [114] have proved that a finitely generated fully residually free group is a subgroup of a group built up in the following manner. A group is commutative transitive or CT if commuting is transitive on non-identity elements. Definition 2.30 Let G = 1 be a commutative transitive group. Let u ∈ G \ {1} and let M = ZG (u). Let B = 1 be a torsion free abelian group. Then G(u, B) = G, B ; rel(G), rel(B), [B, M ] = 1

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is the B-extension of the centralizer M of u in G. If B = t ; is infinite cyclic, then G(u, B) = G, t ; rel(G), t−1 zt = z, for all z ∈ M is the free rank one extension of the centralizer M of u in G. Observe that if B = 1 is an arbitrary torsion-free abelian group, then G(u, B) is a direct union of free rank one extensions of centralizers. One can prove that if G = 1 is fully residually free, then so is every free rank one extension of a centralizer of G. From this it follows that if B = 1 is any torsion-free abelian group and u ∈ G \ {1}, then G(u, B) is locally fully residually free. In the special case where B is residually Z, in particular if B is free abelian, then G(u, B) is fully residually free. Myasnikov and Remeslennikov then prove that a finitely generated fully residually free group must be a subgroup of a group built up from a free group by iterated extensions of centralizers. Theorem 2.31 A finitely generated fully residually free group is in the class of groups formed by taking iterated extensions of centralizers starting with a free group. The concept of a centralizer extension actually has its origins in the paper of Baumslag [6] mentioned earlier where he proved that each Sg , with g ≥ 2, is residually free. Baumslag observed that each Sg , with g ≥ 2, embeds in S2 and residual freeness is inherited by subgroups so it suffices to show that S2 is residually free. He actually showed more. If F is a nonabelian free group and u ∈ F is a nontrivial element which is neither primitive nor a proper power then the Baumslag double K given by K = F  F ; u = u is residually free. To prove this he proceeded by embedding K in the free rank one extension of centralizers H = F, t ; t−1 ut = u by K = F, t−1 F t . The group H is then residually free and hence K is residually free. A limit group of rank d is a limit group built up starting with a free group by d extensions of centralizers. This is fully residually free and hence a limit group. Similar to our plainarity approach this provides an inductive treatment of limit groups. This was used by Fine, Gaglione, Myasnikov, Rosenberger and Spellman [47] to classify the fully residually free groups of small rank. Fully residually free groups must be commutative transitive (CT) and CSA. CSA stands for conjugately separated abelian and signifies that maximal abelian subgroups are malnormal. CSA implies commutative transitive but not conversely in general. However they are equivalent in the presence of full residual freeness. Fine, Myasnikov, grosse Rebel and Rosenberger [57] recently gave a classification of the one-relator CT and one-relator CSA groups building upon previous results

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of Gildenhuys, Kharlampovich and Myasnikov [73]. This classification requires an additional class of groups introduced earlier by Fine and Rosenberger [62] and independently by Cohen and Lustig [39] in connection with very small group actions on R-trees. Definition 2.32 A group G is a restricted Gromov group or RG-group if for any g, h ∈ G then either the subgroup g, h is cyclic or there exists a positive integer k with g k = 1 = hk such that g k , hk = g k  hk the free product of g k and hk . It is known that torsion-free hyperbolic groups are RG (see [62]). The classification is then given in the following three theorems that tie together for one-relator groups the concepts of CT, CSA, RG and the existence of Baumslag-Solitar subgroups. Theorem 2.33 ([57, 60]) Let G be a one-relator group with torsion. Then the following are equivalent: (1) G is CSA. (2) G is RG. (3) G does not contain a copy of the infinite dihedral group x, y ; x2 = y 2 = 1 . (4) G satisfies Lyndon property LZ, that is, if x2 y 2 z 2 = 1 in G then the subgroup x, y, z is cyclic. For torsion-free one-relator groups the situation is different. A torsion-free onerelator group fails to be a CSA group if and only if it contains a copy of some Baumslag-Solitar group B1,n with n = 1 or a copy of the direct product of a free group of rank 2 and an infinite cyclic group F2 × Z (see [57]). In [57] it was proved that a torsion-free one-relator group fails to be an RG-group if and only if it contains a copy of one of the Baumslag-Solitar groups B1,n with n = 0. Recall that B1,1 is a free abelian group of rank 2. It follows that if G is a torsion-free one-relator group which does not contain a free abelian subgroup of rank 2 then the following are equivalent. (1) G is a CSA group; (2) G is an RG-group; (3) G does not contain a copy of some B1,n with n ∈ / {−1, 0, 1}. Further if G is a torsion-free one-relator group then G is CT if and only if it does not contain a copy of F2 × Z or a copy of the Klein bottle group B1,−1 . Combining all these we get the following results. Theorem 2.34 ([57]) Let G be a torsion-free one-relator group which does not contain a copy of Z × Z = B1,1 . Then the following are equivalent: (1) G is CSA; (2) G is RG;

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(3) G does not contain a copy of one of the Baumslag-Solitar groups B1,m = x, y : yxy −1 = xm with m ∈ Z \ {−1, 0, 1}. Notice that since one-relator groups with torsion are commutative transitive, this fact together with Theorem 2.34 provides the total classification of one-relator commutative transitive groups. We remark that these results are very much related to Gersten’s conjecture that a torsion-free one-relator group without BaumslagSolitar subgroups is hyperbolic.

3

Residual Finiteness and the Baumslag Conjecture

In 1967 in [10] Gilbert Baumslag conjectured that all one-relator groups with torsion are residually finite. A one-relator group with torsion has the form G = a1 , . . . , an ; Rm = 1 where R is a freely reduced, cyclically reduced word in the free group on a1 , . . . , an and m > 1. Further all the elements of finite order are conjugate to an element of the cyclic group R . There were partial results on the conjecture by Allenby, Mosner and Tang (see [3]) using particular forms of the relator however the full conjecture was settled affirmatively by Wise in 2009 with a beautiful geometrically inspired proof using what he called cube complex theory. The details of his proof can be found in his monograph [143]. There is also a nice summary of his methods in [25] that can be found on the internet in a series of lectures on Baumslag’s work. In 2007 Baumslag, Miller and Troeger [16] gave an example of a one-relator group (necessarily torsion-free) that is not residually finite showing further the complexity of one-relator group theory. In particular they proved Theorem 3.1 Let G = a1 , . . . , an , . . . ; r = 1 with n ≥ 2. Suppose that w is any element in the free group on a1 , . . . , an which does not commute with r. Then the group w G(r, w) = a1 , . . . , an ; rr = r2 is a one-relator group with the same finite images as G. Further r = 1 in G(r, w) and r is contained in every subgroup of finite index in G(r, w). Therefore G(r, w) is not residually finite. For the remainder of the section we give a brief overview of Wise’s methods. The main theorem is: Theorem 3.2 If G is a one-relator group with torsion, then G is residually finite. His method of proof involves what Wise calls special cube complexes combined with the Magnus breakdown of a one-relator group. This allows for an induction on what he calls a quasi-convex hierarchy. Moldavanskii (see [107,113]) proved the following result that we will explore again in talking about plainarity.

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Theorem 3.3 Let G be a noncyclic one-relator group. Then G is an HNN extension of a one-relator group with a shorter relator. Moldavanskii’s result provides another proof of Magnus’s Freiheitssatz and provides an inductive hierarchy to build up one-relator groups from one-relator groups with shorter relators using HNN extensions. Recall that from the Freiheitssatz subsets of the generators in a one-relator group freely generate free groups. These are called Magnus subgroups. The first part of Wise’s proof uses a quasi-convex hierarchy. (See [75] for a discussion of quasi-convex subgroups of word hyperbolic groups.) A group G has a length k quasi-convex hierarchy for some finite k if it is constructed in the following manner. The trivial group has a length 0 quasi-convex hierarchy. If k ≥ 1 G then G has a length k quasi-convex hierarchy if G is either a free product with amalgamation A C B or an HNN extension G = A, t ; t−1 Ct = C of groups with k − 1 length hierarchies and C is finitely generated and quasi-convex relative to the word metric in G. A special cube complex is a cube complex that has properties that can be viewed as a generalized version of graphs. A group G is virtually special if it has a finite index subgroup isomorphic to the fundamental group of a special cube complex. Wise then proves Theorem 3.4 If G is a hyperbolic group and has a quasiconvex hierarchy then G is virtually special. Theorem 3.5 If G is a hyperbolic virtually special group then G is residually finite. To complete the proof of the residual finiteness conjecture what must be shown is that a one-relator group with torsion is hyperbolic and admits a quasiconvex hierarchy. It was well-known that one-relator groups with torsion are hyperbolic. Being word hyperbolic is implied by having a Dehn presentation. The B. B. Newman Spelling Theorem (see [115]) provides such a Dehn presentation for a one-relator group with torsion. The quasiconvex hierarchy is developed from the Moldavanskii breakdown by showing that in the case of a one-relator group with torsion the associated Magnus subgroups are quasiconvex in the word metric on the group. 3.1

The Coherence of One-Relator Groups with Torsion

It was conjectured by Baumslag that one-relator groups with torsion are coherent. This main result was proved in 2005 by Wise [142] again using geometric methods. We refer to Wise’s paper [142] for more details and an explanation of the extended Hanna Neumann conjecture. An alternative proof was recently given by Louder and Wilton (see [104, 105]). We thank the referee for pointing this out. Theorem 3.6 Let G = a1 , . . . , an ; W m = 1 . Then if m ≥ 4 G is coherent. G is also coherent if m ≥ 2 if we assume the extended Hanna Neumann conjecture.

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An important advance, of a rather different kind, is contained in the paper by Feighn and Handel [46], who proved that a finitely generated group which is an extension of a locally free group by an infinite cyclic group is coherent. This is related to the virtually free-by-cyclic conjecture that we discuss in the next section. 3.2

The Virtually Free By Cyclic Conjectures and the BS Conjectures

Very little is known of the general structure of one-relator groups without additional restrictions. However in a series of papers [23], [14] there has been work on the following wide ranging strong conjecture which implies the residual finiteness property. (See also [100].) Conjecture 3.7 Every one-relator group with torsion is virtually free-by-cyclic, that is, contains a subgroup of finite index which is a cyclic extension of a free group. Further, the conjecture may be true for a wider class of one-relator groups. Each surface group is free-by-cyclic and we believe that each Baumslag double is virtually free-by-cyclic. Since a finitely generated virtually free-by-cyclic group is residually finite [7], the residual finiteness theorem of Wise is a consequence of the above. Moreover finitely generated free-by-cyclic groups are also coherent (Feighn and Handel [46]), that is, their finitely generated subgroups are finitely presented. This then impacts on Wise’s result that one-relator groups with torsion are coherent [8]. A great deal of effort has gone into proving these conjectures. Now a one-relator group with torsion contains a subgroup of finite index which is torsion-free ([86] and [68]). Our hope is that one of these torsion-free subgroups is an infinite cyclic extension of a free group. Using a combination of Reidemeister-Schreier methods and computation, Baumslag and Troeger [23] show how to construct the virtually free-by-cyclic structure for an expicit Fuchsian group with torsion. In [14] a study of the virtually free-by-cyclic structure of cyclically pinched onerelator groups shows that it is rather well behaved. We let VFC denote the class of virtually free-by-cyclic groups. Then: Theorem 3.8 The class VFC is closed under taking subgroups and free products. Free-by-cyclic groups and virtually free-by-cyclic groups arise in many different contexts. In particular the fundamental groups of all orientable surfaces of genus g ≥ 2 and nonorientable of genus g ≥ 3 are free-by-cyclic. It follows that all finitely generated Fuchsian groups are virtually free-by-cyclic. In the same spirit a result of Howie [78] gives sufficient conditions for the fundamental group of a 2-complex to be free-by-cyclic in terms of a Morse function. In [14] several results were proved concerning the free-by-cylic structure of both cyclically pinched one-relator groups and conjugacy pinched one-relator groups. Theorem 3.9 Suppose that G = A  B is a cyclically pinched one-relator group. U =V

If U ∈ / [A, A] and V ∈ / [B.B] then G is free-by-cyclic.

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Theorem 3.10 Suppose that G = F, t ; t−1 U t = V with U, V ∈ / [F, F ] is a conjugacy pinched one-relator group. If either U [F, F ] = V [F, F ] or U, V are linearly independent modulo [F, F ] then G is free-by-cyclic. These two results deal with elements not in the derived group. We can make progress when they are in the derived group in the case of Baumslag doubles. In particular Theorem 3.11 Baumslag doubles are virtually free-by-abelian. That is G=F

 F U =U

and

H = F, t ; t−1 U t = U

are virtually free-by-abelian. There are also technical sufficient conditions so that doubles are actually virtually free-by-cyclic (see [14]). 3.3

Amalgam Decompositions of One-Relator Groups

Baumslag and Shalen [21] proved the following. Theorem 3.12 Let G be a one-relator group with at least 3 generators. Then G has a nontrivial free product with amalgamation decomposition G = A H B where A and B are finitely generated and H is finitely generated. An alternative proof in the more general case of one-relator products of cyclics with torsion was given by Fine, Levin and Rosenberger [55] using the dimension of the character variety (see also [64]). Recently Benyash-Krivets [24] extended this to show that all noncyclic one-relator groups with torsion are nontrivial free products with amalgamation. Unfortunately in these decomposition results very little is known about the exact nature of the factors and this would have to be studied to gain more information about one-relator groups. Fine and Peluso [58] call a free product with amalgamation decomposition of a group G with finitely generated factors a BaumslagShalen Decomposition. They conjecture that a one-relator group with at least three generators has a Baumslag-Shalen decomposition where the factors are either free groups or one-relator groups with free amalgamated subgroups. They prove this conjecture up to homology and prove it exactly for several special cases [58]. As mentioned in Section 3, Baumslag and Shalen [21], Fine, Levin and Rosenberger [55] and finally Benyash-Krivets [24], showed that all noncyclic one-relator groups with torsion are nontrivial free products with amalgamation. This holds also for torsion-free one-relator groups with at least 3 generators [21]. Unfortunately in these decomposition results very little is known about the exact nature of the factors. Fine and Peluso make several conjectures concerning the BS-decomposition of one-relator groups (see [58]).

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Conjecture 3.13 (BS Conjecture 1) Let G be a one-relator group with at least three generators. Then G has a BS-decomposition G=AB H

where the factors A and B are either one-relator groups or free groups and H is free. Conjecture 3.14 (BS Conjecture 2 (The Strong BS Conjecture)) Let G be a one-relator group. Then if G=AB H

is a BS-decomposition of G with H free then the factors A and B are either onerelator groups or free groups. The strong BS Conjecture was proved up to homology in [58]. Related to these conjectures is the general question of the subgroup structure of a one-relator group. The embedding theorems in G. Baumslag [12] provide rather different ways of embedding infinitely related groups explicitly in finitely presented groups. Whether such techniques can be used in studying the subgroup structure of one-relator groups is worth investigating further. The existence of these amalgam decompositions for one-relator groups can be used to prove that many of them are SQ-universal. From a result of Sacerdote and Schupp [125] a torsion-free one-relator group with at least three generators is SQuniversal. From the Baumslag-Shalen decomposition theorem above together with a result of Lossov [103] it follows that a one-relator group with torsion and at least two generators is SQ-universal. A recent result of Benyash-Krivets [24] showed that all noncyclic one-relator groups with torsion are nontrivial free products with amalgamation. Combining this with the result of Lossov gives that any noncyclic one-relator group with torsion is SQ-universal. This last result also can be deduced easily from the SQ-universality arguments for one-relator quotients of free products of cyclic groups given in [64]. Theorem 3.15 Let G be a torsion-free one-relator group with at least three generators. Then G is SQ-universal. Theorem 3.16 Let G be a noncyclic one-relator group with torsion. Then G is SQ-universal. This was recently extended by Button and Kropholler [35] who proved that a noncyclic two-generator one-relator group is SQ-universal or isomorphic to some B(1, m).

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Some Problems Concerning One-Relator Groups

In this section we discuss a collection of problems concerning one-relator groups that we propose can be attacked via the plainarity approach. We do not claim this list is comprehensive and there are other important open problems in one-relator group theory. Some work in this direction has already been done on the virtually free-by-cyclic conjecture [14,23]. In the subsequent subsections we comment on the status of the problems, the past work and potential avenues of attack. 4.1

The Standard Decision Problems

Combinatorial group theory has always been concerned with the three major decision problems of Max Dehn; the word problem, the conjugacy problem and the isomorphism problem. Using the Freiheitssatz and the Magnus breakdown, Magnus was able to prove (see [109]) that the word problem and generalized word problem are solvable for all one-relator groups. However the conjugacy problem has proved to be quite difficult. Lipschutz [101] proved that the conjugacy problem is solvable for cyclically pinched one-relator groups. One-relator groups with torsion are hyperbolic. Hence they have solvable conjugacy problem. This was proved first by B. B. Newman [115], by means of his so-called spelling theorem. Here we note that every one-relator group with torsion has a torsion-free, fully invariant subgroup of finite index (see [86]). A considerable effort over many years was expended by Juhasz [80–82], who attempted to use ideas from small cancellation theory, but with only partial success. Other partial results have been obtained in particular by Ivanov and Schupp [79]. In general the conjugacy problem remains open. Theorem 4.1 ([101, 115]) The conjugacy problem is solvable for both cyclically pinched one-relator groups and for one-relator groups with torsion. The isomorphism problem is of course the most difficult of the decision problems. For one-relator groups there are some major results. Using Nielsen reduction in free products with amalgamation, Rosenberger [124] proved that the isomorphism problem is solvable for cyclically pinched one-relator groups. Theorem 4.2 ([124]) The isomorphism problem is solvable for cyclically pinched one-relator groups. Pride did the same for 2-generator one-relator groups with torsion [117]. Theorem 4.3 The isomorphism problem is solvable for 2-generator one-relator groups with torsion. Dahmani and Guiradel [41] proved that the isomorphism problem for hyperbolic groups with torsion is solvable. Sela had earlier proved the solvability of the isomorphism problem for torsion-free hyperbolic groups [126]. One-relator groups with torsion are hyperbolic and therefore it follows that all one-relator groups with torsion have a solvable isomorphism problem.

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Theorem 4.4 ([41]) The isomorphism problem is solvable for one-relator groups with torsion. In [8] G. Baumslag introduced the following class of groups: Gm,n = a, b, t ; a−1 = [bm , a][bn , t]

with m, n ≥ 1.

He then showed that these groups are all parafree. However Magnus and Chandler [37] in their History of Combinatorial Group Theory mention these groups as an example of the difficulty of the isomorphism problem for one-relator groups. Up until 1993 there was no proof showing that any of the groups Gm,n are non-isomorphic. Liriano [102] using representations of Gm,n into P SL2 (pk ) showed that G1,1 and G30,30 are non-isomorphic. Subsequently in 1997 Fine, Rosenberger and Stille [67] using Nielsen cancellation methods showed that the isomorphism problem is solvable for the subclass Gn,1 . Further it can be decided algorithmically whether or not an arbitrary one-relator group is isomorphic to Gn,1 . In particular Theorem 4.5 ([66]) Let n be a natural number and Gn,1 be the group defined above. Then: (1) The isomorphism problem for Gn,1 is solvable, that is, it can be decided algorithmically in finitely many steps whether or not an arbitrary one-relator group is isomorphic to Gn,1 . (2) Gn,1 is not isomorphic to G1,1 for n ≥ 2. (3) If p, q are primes then Gp,1 ∼ = Gq,1

if and only if

p = q.

Further for all m the group Gm,1 is hopfian and then every automorphism of Gm,1 is induced by an automorphism of the free group F on a, b, t. In addition the automorphism group of Gm,1 is finitely generated. Sela [126] has proved, among other things, that torsion-free hyperbolic groups are hopfian A subgroup of finite index in a hyperbolic group is again hyperbolic. The hopficity of torsion-free hyperbolic groups then allows one to deduce that every onerelator group with torsion is also hopfian (see [9]). The existence of a fully invariant torsion-free subgroup of finite index in one-relator groups with torsion, reduces the isomorphism problem for one-relator groups with torsion to extension theory and knowledge of the finite subgroups of the automorphism groups of torsion-free hyperbolic groups. Two elements of a free group are conjugate if and only if they are conjugate in every nilpotent factor group. Since the conjugacy problem is solvable for finitely generated nilpotent groups, this might provide an approach to the conjugacy problem for one-relator parafree groups. Such an approach cannot work for all finitely presented, residually nilpotent groups in view of the results of G. Baumslag [6]. This focuses attention on one-relator parafree groups and the question as to whether all such groups are hyperbolic, which would have a solution of the conjugacy problem as a byproduct. Now a finitely presented group is hyperbolic if and only if it

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satisfies a linear isoperimetric function. The very form of the relator of a parafree one-relator group suggests that it might be possible to compute its isoperimetric function directly. The computation of isoperimetric functions of one-relator groups of plainarity 1 breaks down into several cases and many of them can be handled directly. The conjugacy problem for one-relator groups of plainarity 1 can also be handled directly and appears to be very promising. Baumslag and Cleary [13] recently introduced three new families of non-free parafree groups and discuss limitations to a natural procedure for distinguishing these groups from one another. 4.2

Algorithmic Problems for One-Relator Groups

There are many problems in free groups that can be algorithmically solved, for example the word problem and generalized word problem and the problem of determining the intersection of two finitely generated subgroups. Using infinite words many of these algorithmic methods can be extended to finitely generated fully residually free groups (see [96]). Given the closeness of one-relator groups to free groups it is of interest to study algorithmic questions in one-relator groups. Here we actually mean algorithmic problems for finitely generated one-relator presentations. A one-relator group can of course have a non-one-relator presentation. For algorithmic problems non-one-ralator presentations does not present a problem. If we know that G has a one-relator presentation then we can systematically enumerate the presentations for G using Tietze transformations until we find a one-relator presentation. We thank the referee for reminding us of this fact. Many if not most algorithmic problems about one-relator groups in general (excluding surface groups) have yet to be investigated. Here are a few whose solutions are known in free groups that have not been touched on before. (1) Is there an algorithm whereby one can decide whether or not an element in a one-relator group is a commutator or a proper power? (2) Is there an algorithm to decide whether or not a given element in a given one-relator group lies in a given finitely generated subgroup? (3) Is there an algorithm to decide whether or not any given finitely generated subgroup of a one-relator group is normal or malnormal? These and other problems apply in particular to parafree one-relator groups. 4.3

Linearity and Linearity Conditions for One-Relator Groups

In general it is not known which one-relator groups are linear. From the results of Poincar´e which were formalized by Fricke and Klein [69] it follows that surface groups are linear. This has been generalized in several ways. Wehrfritz [138] showed that a cyclically pinched one-relator group where neither U nor V are proper powers has a faithful representation over a commutative field. This was improved upon by Shalen [134] and generalized by Fine and Rosenberger [61] to show that cyclically pinched one-relator groups with neither U nor V proper powers and more generally

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all groups of F-type have faithful representations in P SL2 (C). A group of Ftype is a group that has a presentations of the form G = a1 , . . . , an ; ae11 = · · · = aenn = 1, U = V where ei = 0 or ei ≥ 2 for i = 1, . . . , n and U = U (a1 , . . . , ap ), V = V (ap+1 , . . . , an ) are nontrivial cyclically reduced words in the free product of cyclics on the generators that they involve and where neither U nor V are proper powers. This shows in particular that each group in this class is residually finite and Hopfian. Further they satisfy the Tits alternative, that is they either contain a nonabelian free group or are virtually solvable and further they are virtually torsion-free (from a result of Selberg [133]). In general if a group G has a presentation a1 , . . . , an ; ae11 = · · · = aenn = 1, R1m1 , R2m2 , . . . , Rkmk where ei = 0 or ei ≥ 2 for i = 1, . . . , n and mj ≥ 1 for j = 1, . . . , k, and each Rj is not a proper power then an essential representation of G is a representation ρ : G → H with H a linear group such that ρ(ai ) has exact order ei for each i and ρ(Rj ) has exact order mj . A result of Mendelsohn and Ree [111] which was generalized by Fine, Howie and Rosenberger [51] shows that a one-relator product of cyclics G = a1 , . . . , an ; ae11 = · · · = aenn = 1, Rm = 1 where ei = 0 or ei ≥ 2 for i = 1, . . . , n has an essential representation whenever m ≥ 2. This had been done earlier for generalized triangle groups by Baumslag, Morgan and Shalen [17]. From this it follows that many one-relator products of cyclics and hence many one-relator groups share a large number of linearity properties with linear groups even though they may not be linear (see the book [64] for an extensive discussion of this). We mention a few of these concerning cyclically pinched one-relator groups and one-relator groups with torsion. Theorem 4.6 Let G be a cyclically pinched one-relator group with either U or V not a proper power. Then (1) G has faithful representations in P SL2 (C) and P SL2 (R); (2) G is residually finite; (3) G is hopfian. Theorem 4.7 Let G be a one-relator group with torsion. Then (1) G has an essential representation in P SL2 (C); (2) G is virtually torsion-free. Many of these linearity results can be extended to a wider class of groups called groups of F-type, as above, which generalize both cyclically pinched one-relator groups and Fuchsian groups. These were introduced by Fine and Rosenberger in [61] and a complete description and discussion is in the book [64].

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The proof that a group is linear is a daunting task in general. The most general approach is Lubotzky’s remarkable characterisation of finitely generated linear groups [106]. 4.4

The Hyperbolicity and Automaticity Problem

The classification theorems of the previous section are related to the Gersten conjecture that a torsion-free one-relator group is hyperbolic if and only if it does not contain any Baumslag-Solitar group as a subgroup. It is known that one-relator groups with torsion are hyperbolic (see [15]). Gersten has also asked whether a finitely generated one-relator group whose abelian subgroups are cyclic, is hyperbolic. This is one of the most intriguing problems about one-relator groups. B. B. Newman [115] proved that the abelian subgroups of a one-relator group are either cyclic, locally cyclic of a particularly restricted form or free abelian of torsion-free rank two. The abelian subgroups of hyperbolic groups are all finitely generated of rank at most one (see [126]). Reconciling the structure of a one-relator group with this restriction on its abelian subgroups with that of a hyperbolic group is an extremely elusive endeavour. Some light may well be thrown on this problem by restricting to one-relator groups of plainarity 1. It is known that certain cyclically pinched one-relator groups and conjugacy pinched one-relator groups are hyperbolic. More generally, hyperbolicity is preserved under special amalgam constructions. Bestvina and Feighn [26] have shown that an amalgam of two hyperbolic groups over a quasiconvex cyclic subgroup that is malnormal in at least one factor is still hyperbolic. This implies that certain cyclically pinched one-relator groups are hyperbolic. Kharlampovich and Myasnikov [90] have a more general result that the amalgam of two hyperbolic groups is again hyperbolic whenever one of the amalgamated subgroups is quasiconvex and malnormal in its respective factor. Related results were proved by Juhasz and Rosenberger [83] for cyclically pinched one-relator groups and groups of F-type. Theorem 4.8 If H1 , H2 are hyperbolic and H1 ∩ H2 = H is a quasiconvex subgroup, malnormal in either H1 or H2 , then the amalgamated product H1 H H2 is hyperbolic. In particular if W1 , W2 are elements of infinite order in H1 and H2 respectively and neither is a proper power then H1



W1 =W2

H2

is hyperbolic. Since finitely generated subgroups of free groups are quasiconvex it follows that if A and B are free groups and A ∩ B = H is a finitely generated subgroup malnormal in either A or B then the amalgamated product A H B is hyperbolic. Both Kharlampovich and Myasnikov [90] and Bestvina and Feighn [26] have unrelated results concerning the hyperbolicity of HNN extensions. The first result can be specialized to many conjugacy pinched one-relator groups. A separated

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HNN-extension is an HNN group K = G, t : t−1 At = B where the associated subgroups have the property that g −1 Ag ∩ B = {1} for any g ∈ G. Theorem 4.9 ([90]) A separated HNN extension G, t : t−1 At = B of a hyperbolic group G is hyperbolic if the associated subgroups A and B are quasiconvex in G and at least one is malnormal. Theorem 4.10 ([26]) Let F be a finitely generated free group with basis a1 , . . . , an . Let f ∈ Aut(F ). Then the mapping torus M = a1 , . . . , an , t ; t−1 ai t = f (ai ), i = 1, . . . , n is hyperbolic if and only if the automorphism f has no non-trivial periodic conjugacy classes. This also leads to another conjecture of Wise that a one-relator group is automatic if and only if it does not contain a subgroup Bk,m with |k| =  |m|. This conjecture was proved to be false (see [72]). We have from the above discussion a large collection of automatic one-relator groups. The general problem as to which one-relator groups are automatic is unknown and an extremely interesting one. The study of the possible isoperimetric functions for one-relator groups is particularly relevant. Again a start on this problem can be made by considering one-relator groups of plainarity 1. Questions of automaticity can also be linked to the study of parafree groups (see [13]). Stallings [136] has proved that if the derived factor group of a residually nilpotent group G is free abelian and if its second integral homology group is zero, then G is parafree. This suggests the possibility of characterising, for example, parafree one-relator groups. Parafree groups arise naturally in homotopy theory (see the lecture notes of Bousfield and Kan [30] and also the work of Brown and Dror [32]). Various attempts to prove that all finitely generated parafree groups have finitely generated second homology groups and more generally that they are finitely related have been unsuccessful (see [77]). The diversity of presentations that give rise to such groups mark them as a fascinating generalization of free groups. As an example of this diversity, here are two examples of one-relator parafree groups: G = a, b, c ; a2 b3 c5

and

H = a, b, c ; a = [a, b][c, b] .

They are probably not isomorphic – neither of them is free. Both groups are automatic and the first of them is hyperbolic; whether the second one is hyperbolic is still unknown. These remarks again underline the difficulty of the isomorphism problem for one-relator groups. One of the difficulties here is to reconcile conditions imposed on the lower central series of a group with the geodesics in its Cayley graph.

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Baumslag, Fine, Rosenberger: One-relator groups The Residual Nilpotence and Residual Solvability Problem

As we have discussed, residual freeness plays a critical role in modern infinite group theory. Since free groups are residually nilpotent we have that residually free groups are also residually nilpotent. Therefore the general question of residual nilpotence and residual solvability of one-relator groups is an interesting one and a problem that may be approached through plainarity. It is easy to prove that there is no algorithm whereby one can decide whether or not any finitely presented group is residually nilpotent or even residually solvable. It is hard to believe that these same problems are undecidable for one-relator groups. However not too much is known here. An example of a one-relator group which is not residually solvable is the following group: G = a, b ; (a[a2 , b−1 a2 b])2 . The group G is a one-relator group with torsion. G is not residually solvable. This follows from the fact that the given defining relation can be re-expressed in the form u = u , where u is in the derived group of the normal closure of u. In other words one can discern from the very defining relation of G that it is not residually solvable. This discussion suggests a number of ways of settling the problem of residual solvability for one-relator groups. For example, if a defining relation cannot be re-expressed in the form above, is the given group residually solvable? The most general result in this area is the proof that positive one-relator groups are residually solvable ([9]). Here a one-relator group is termed positive if its defining relator is a positive word, that is, involves only positive powers of the given generators. Another example of a family of one-relator groups which are residually solvable are the groups which can be presented in the form G = X ; [u, v] where here u and v are positive words ([11]). The question arises whether all onerelator groups of the form G = X ; [u, v] are residually solvable, where now u and v are abitrary words. One can prove a lot more than residual solvability in the event that G = Y, t ; [u, t] , where u is a Y -word which is not a proper power. In this case G turns out to be fully residually free (see [48]). The proof of this theorem can be applied also to certain completions of free groups and so has applications which go beyond the theory of one-relator groups. It should be noted that since free groups are residually torsion-free nilpotent, so are the groups G which can be presented in the form G = Y, t ; [u, t] . 4.6

The Algebraic Geometry of One-Relator Groups

Algebraic geometry over groups, especially over free groups, has been one of the major developments in combinatorial group theory over the past decade and half. Introduced originally in [18] and continued in many other places it has provided

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the main tools for the solution of the Tarski problems (see [48]). However except for surface groups very little is known in general about the algebraic geometry of one-relator groups. The nature of the algebraic geometry of one-relator groups then is a very important question. Here the algebraic geometry of one-relator groups of plainarity 1 should be quite close to that of free groups. Algebraic geometry depends on a group being equationally Noetherian. As already mentioned, Guba [76] has proved that all linear groups are equationally Noetherian. Since free groups are linear, this again focusses attention on the linearity of one-relator groups and in particular parafree one-relator groups. From the results of Wehrfritz [138], Shalen [134] and Fine and Rosenberger (see [64]) it follows that many parafree one-relator groups are linear and hence equationally Noetherian. The paper [20] contains examples of one-relator groups which are not equationally Noetherian. 4.7

Discerning Properties of One-Relator Groups from the Form of the Relator

A quadratic relator R is one in which each generator involved either appears twice with exponent ±1 or once with exponent 1 and once with exponent -1. The relator R in the standard one-relator presentations for both orientable and nonorientable surface groups are quadratic relators. Much of the properties, especially linear properties, can be deduced from this fact. This raises the general question of how much information can be gleaned about a one-relator group solely on the general form of the relator analogous to the quadratic case. The case of cyclically pinched one-relator groups and conjugacy pinched one-relator groups can be considered a generalization of this. This question takes two forms; specific format for the relator, such as being quadratic and length of the relator. Wise (see [142]) has effectively used the length of the relator to handle the residual finiteness conjecture. 4.8

The Nature of the Automorphism Group of a One-Relator group

Suppose that G = F ; R is a finitely presented group with F  a finitely generated free group on the generators F . An automorphsim α : G → G is tame if it is induced or lifts to an automorphism on F  . If each automorphism of G is tame we say that the automorphism group Aut(G) is tame. In [135] Shpilrain gives a survey of some of the known general results on tame automorphisms and tame automorphism groups. If G is a surface group a result of Zieschang [144] which was improved upon by Rosenberger [121] shows that G has only one Nielsen class of minimal generating systems. An easy consequence of this is that Aut(G) is tame. Rosenberger uses the term almost quasifree for a group which has a tame automorphism group. If G = F ; R is almost quasifree and in addition each automorphism of F induces an automorphism of G, G is called quasifree. Rosenberger observed that a non-cyclic one-relator group is quasifree only if it has a presentation a, b ; [a, b]n = 1 for n ≥ 1. This is a Fuchsian group if n ≥ 2 or isomorphic to a free abelian group of rank 2 if n = 1. The result of Fine, Rosenberger and Stille (Theorem 4.5) shows that the automorphism groups

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Aut(Gn,1 ) are tame where the Gn,1 are the parafree groups introduced by Baumslag in [8] (see above). From Rosenberger’s result on cyclically pinched one-relator groups [124] it follows that the automorphism group of a cyclically pinched one-relator group is tame (see also [122, 123]). However very little is known about the general structure of the automorphism group of a one-relator group. The special case of one-relator groups of plainarity one seems doable. In another direction, extending the tameness result for surface groups, Fine, Kharlampovich, Myasnikov, Remeslennikov and Rosenberger [53] proved that fully residually free groups and elementary free groups have tame automorphism groups. In particular they prove the following using the JSJ decomposition of limit groups together with results on the isomorphism problem for fully residually free groups [34]. Theorem 4.11 The automorphism group Aut(G) of a finitely generated freely indecomposable fully residually free group G is tame. From this it follows that the automorphism group of a finitely generated freely indecomposable elementary free group G is tame. We note that the converse of Theorem 4.11 is false. That is there do exist groups (in fact hyperbolic groups) where every automorphism is tame but which are not fully residually free (see [95]). An interesting fact in the study of automorphisms is a result of Kapovich and Schupp [85] that the isomorphism class of a one-relator group G = X ; R is generically (which can be thought as being of asymptotic density one) the class G = X ; τ (R) where τ runs over the set of Whitehead automorphisms. We refer the reader to the paper of Kapovich and Schupp [85] for a further explanation of generic properties and asymptotic density since it is rather far afield from the present discussion. 4.9

Some Extensions to Two-Relator Groups

Groups with two defining relators constitute a class which has yet to be investigated in any detail. A type of Freiheitssatz for certain two-relator groups was proved by I. Anshel (see [4] and [27]). The work by Elvira Rapaport [118] on the Poincar´e conjecture seems to point the way to a fruitful exploration of certain other tworelator groups. We recall Rapaport’s theorem. To this end, let G = a1 , b1 , . . . , ak , bk ; [a1 , b1 ] . . . [ak , bk ], [a1 , w] , where w is an arbitrary word in the given generators. Then Rapaport proves that G is torsion-free. It should be pointed out that a non-trivial commutator in a free group is never a proper power (Karrass, Magnus and Solitar [86]). This implies that a one-relator group, whose defining relator is a commutator, is torsion-free [86]. Rapaport makes use of this fact in her proof. Now surface groups are fully residually free. It seems likely that this observation can be used to find an essentially different proof of Rapaport’s theorem. It also hints at possible extensions to many groups defined by two relations.

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5

149

The Magnus Breakdown and Plainarity

The primary general result on one-relator groups is the Freiheitssatz or Independence Theorem originally proved by Magnus in 1929 [108]. Theorem 5.1 Let G = x1 , . . . , xn ; R where R is a cyclically reduced word in the free group on x1 , . . . , xn which involves all the generators. Then the subgroup generated by x1 , . . . , xn−1 is free on these generators. There are many good discussions of the Freiheitssatz and its generalizations. The articles by Baumslag [10] and Fine and Rosenberger [63] provide a wealth of information. In the course of proving the Freiheitssatz, Magnus developed a technique for handling one-relator groups in general. He proved that a one-relator group can be broken down into amalgamated products of simpler one-relator groups, thereby providing an inductive mechanism for handling them. We call this whole procedure the Magnus Breakdown. An in-depth focus on Magnus’s approach will be a primary tool, though not the only one, in any exploration of a one-relator group. Usually the first step in an induction of the kind alluded to above with what we will now term plain one-relator groups, in accordance with the following definition. Definition 5.2 Let G = a, . . . ; r be a one-relator group. The group G is said to be plain if r is a power of a primitive element in the underlying free group on the given generators a, . . . of G. This definition of plain is distinct from the definition in [84]. However this definition does imply the notion of plain defined there. Magnus’s basic idea has been reworked in terms of HNN extensions. In fact Moldavanskii [113] has shown that if G = a, . . . ; r is a one-relator group and if G is not plain, then G can be embedded, in a very simple way, in a closely related one-relator group G0 , which often coincides with G. G0 is an HNN-extension of a second one-relator group H; the length of the given defining relator of H is smaller than the length of r. This procedure gives rise to a series of one-relator groups G0 , G1 , . . . ending up in a plain one-relator group Gd . The length d of such a series depends on a number of choices that are available at almost every step in the production of the series. We say that that G is of plainarity d if it has such a series of length d. Notice that the one-relator groups of plainarity 0 are the plain ones. The plain one-relator groups are either free or the free product of a finite cyclic group and a free group. Thus they can be considered as being well-known. The answers to all of the problems and questions that we consider are known for such plain one-relator groups. The one-relator groups of plainarity 1 are more difficult to investigate than the ones of plainarity 0. It is worthwhile to consider a specific example to illustrate how such groups of plainarity 1 arise and to focus on a few of the problems that we raised earlier in this particular case.

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To this end, let n be a given positive integer and let G = Gn = a, b, c, t ; ([a, b][c, t])n . Then Gn is an HNN-extension of the one-relator group n H = Hn = a0 , b0 , c0 , c1 ; ([a0 , b0 ]c−1 0 c1 ) .

The relator here is primitive in the free group on a0 , b0 , c0 so H is a plain one-relator group and therefore Gn has plainarity 1. In this case for all n ≥ 1 the group Gn is actually a nonelementary Fuchsian group and is therefore linear. It follows that Gn is residually finite and coherent and we can write down all the subgroups of Gn . Further Gn is virtually a surface group and so G is virtually free-by-cyclic. What we do now is show how to prove these facts without recourse to the theory of Fuchsian groups. First of all if n = 1, that is, if G is a surface group, then H is actually free. It then follows from the theory of groups acting on trees, that G is coherent, which demonstrates how useful this approach sometimes is. In order to better understand the structure of G, however, one needs to dig a little deeper. With this in mind, consider the normal closure N in G of H. It is clear that G/N is infinite cyclic on tN . It follows either from Magnus’s original method or from the ReidemeisterSchreier process, that N has the following presentation n N = . . . , a−1 , a0 , a1 , . . . , . . . , b−1 , b0 , b1 , . . . , c0 , . . . , ([ai , bi ]c−1 (i ∈ Z) . i ci+1 )

In the event that n = 1, this implies that N is free on the ai , bi and c0 and that the action of t (by conjugation) on N simply shifts the ai and bi by increasing the subscripts by 1 and conjugates c0 to c0 [b0 , a0 ]. This observation can be used to prove that surface groups are actually residually torsion-free nilpotent and indeed many other one-relator groups as well. This is of course true for any fully residually free one-relator group. In the event that n > 1, N turns out to be the free product of two groups. The first is the free group on the ai , the bi and c0 and the second is the free product of infinitely many cyclic groups of order n, namely the subgroups generated by the elements [ai , bi ]c−1 i ci+1 . On keeping track of the action of t on N , one can then deduce that G is residually finite. If n is not a prime, then G is not residually nilpotent. If one now switches gear and attempts to prove G linear, one needs to find a subgroup of G of finite index which is residually nilpotent. This requires some additional computation. In the event that n = 2, there is a homomorphism of G onto a finite group of order 32 in which the image of [a, b][c, t] is of order 2. The kernel K of this homomorphism has a presentation on 97 generators and 32 defining relations. In general the computation of such presentations requires the use of a computer and it is hard to understand the impact of these presentations. Nonetheless this procedure will constitute an important part of this project. In this special case, it is likely that the method of Wehrfritz in [138] can be modified to prove the linearity of G for every choice of n. It is clear from the above discussion that a complete understanding of one-relator groups of plainarity 1 will be an important step forward. In particular, the proof of the residual finiteness of an

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151

HNN extension depends heavily on the precise nature and “positioning” of the associated subgroups and the associating isomorphism. A detailed investigation may well provide enough information for considerable progress to be made in this direction. The proof of the residual solvability of positive one-relator groups is a simple instance where this kind of knowledge was key. One point of the discussion above is that the complexity of a given one-relator group is often dissipated when going over to a subgroup of finite index and the structure of the group becomes more easily understood. This approach was the impetus for several of the conjectures made by the first author, especially the conjecture that one-relator groups with torsion are residually finite. The recent work of Wise [140, 141] is a clear indicator that some of the mysteries of one-relator groups with torsion will be unravelled by such means. The next example will illustrate what we have in mind. We try to show how to unravel a one-relator group with torsion. Consider the one-relator group G = t, a ; (t2 ata2 ta2 )2 . Next consider the normal closure N in G of a10 and t. Then it is not hard to see that N is of index 10 in G. By simply following the Reidemeister-Schreier method one can show that N can be decomposed into an amalgamated free product of two free groups, one of rank 5 and the other of rank 6. In a little more detail, put ti = ai ta−i (i = 0, 1, . . . , 9),

q = a10

and A = gp(t0 , t1 , t2 , t3 , t4 ),

B = gp(t5 , t6 , t7 , t8 , t9 , q).

Then it turns out that A and B are free on the exhibited generators. Moreover if we now put H = gp(t20 t1 t3 t4 , t0 t21 t2 t4 , t0 t1 t22 t3 , t1 t2 t23 t4 , t0 t2 t3 t24 ) and K = gp((t25 t6 t8 t9 q)−1 , (t5 t26 t7 t9 q)−1 , (t5 t6 t27 t8 q)−1 , (t5 t7 t8 t29 q)−1 , (t6 t7 t28 t9 q −1 )) then H and K are free on the given generators and N is the amalgamated product of A and B with H identified with K, where the first generator of H is identified with the first generator of K, the second generator of H is identified with the second generator of K and so on. This kind of simplification that arises in the subgroups of finite index in one-relator groups with torsion is typical and will be further exploited in the general investigation. We would like to thank the referee for the excellent job done in reviewing this manuscript. The referee’s suggestions greatly improved the paper. References [1] M. Aab and G. Rosenberger, Subgroup separable free products with cyclic amalgamtion, Results in Math. 28 (1995), 185–194. [2] P. Ackermann, B. Fine and G. Rosenberger, Surface groups: Motivating examples in combinatorial group theory, Groups St Andrews 2005, London Math. Soc. Lecture Notes Series 339 (2007), 96–130.

152

Baumslag, Fine, Rosenberger: One-relator groups

[3] R. Allenby, L. E. Mosner, and C. Y. Tang, The residual finiteness of certain onerelator groups, Proc. Amer. Math. Soc. 75 (1) (1980), 8–10. [4] I. Anshel, On two relator groups, in Topology and Combinatorial Group Theory, Springer Lecture Notes in Mathematics 1440 (1990), 1–21. [5] B. Baumslag, Residually free groups, Proc. London Math. Soc. 17 (1967), 402–418. [6] G. Baumslag, On generalised free products, Math. Z. 78 (1962), 423-438. [7] G. Baumslag, Groups with the same lower central sequence as a relatively free group. I. The groups, Trans. Amer. Math. Soc. 129 (1967), 308–321. [8] G. Baumslag, Residually finite one-relator groups, Bull. Amer. Math. Soc. 73 (1967), 618–620. [9] G. Baumslag, Finitely generated cyclic extensions of free groups are residually finite, Bull. Austral. Math. Soc. 5 (1971), 87–94. [10] G. Baumslag, A survey of groups with a single defining relation, Groups St Andrews 1985 (Cambridge University Press, 1986), 30–56. [11] G. Baumslag, Musings on Magnus, in The mathematical legacy of Wilhelm Magnus: groups, geometry and special functions (Brooklyn, NY, 1992), Contemp. Math. 169 (Amer. Math. Soc., Providence, RI, 1994), 99–106. [12] G. Baumslag, Embedding wreath-like products in finitely presented groups, Contemp. Math. 372 (2005), 197–206. [13] G. Baumslag and S. Cleary, Parafree one-relator groups, J. Group Theory 9 (2006), 191–201. [14] G. Baumslag, B. Fine, C. Miller and D. Troeger, Virtual properties of cyclically pinched one-relator groups, Int. J. Alg. Comp. 19 (2009), 1–15. [15] G. Baumslag, S. Gersten, M. Shapiro and H. Short, Automatic groups and amalgams, J. Pure Appl. Algebra 76 (3) (1991), 229–316. [16] G. Baumslag, C. Miller and D. Troeger, Reflections on the residual finiteness of one-relator groups, Groups Geom. Dyn. 1 (2007), 209–219. [17] G. Baumslag, J. Morgan and P. Shalen, Generalized triangle groups, Math. Proc. Camb. Phil. Soc. 102 (1987), 25–31. [18] G. Baumslag, A. G. Myasnikov and V. N. Remeslennikov, Algebraic geometry over groups. I. Ideals and algebraic sets, J. Algebra 219 (1999), 16–79. [19] G. Baumslag, A. G. Myasnikov and V. N. Remeslennikov, Discriminating completions of hyperbolic groups, Geom. Dedicata 92 (2002), 115–143. [20] G. Baumslag, A. G. Myasnikov and V. Romankov, Two theorems on equationally Noetherian groups, J. Algebra 194 (1994), 654–664. [21] G. Baumslag and P. Shalen, Amalgamated products and finitely presented groups, Comment. Math. Helv. 65 (1990), 243–254. [22] G. Baumslag and D. Solitar, Some two-generator one-relator non-hopfian groups, Bull. Amer. Math. Soc. 68 (1962), 199–201. [23] G. Baumslag and D. Troeger, Virtually free-by-cyclic groups I, in Aspects of Infinite Groups (World Scientific Press, 2009). [24] V. V. Benyash-Krivets, Decomposing one-relator products of cyclic groups into free products with amalgamation, Mat. Sb. 189 (1998), 13–26. [25] Bernstein Seminar, Cornell University, Exploring the Works of Gilbert Baumslag – One relator Groups 111 – Wise’s Solution to Baumslag Conjecture, Online. [26] M. Bestvina and M. Feighn, A combination theorem for negatively curved groups, J. Diff. Geom. 35 (1992), 85–101. [27] W. A. Bogley, An identity theorem for multi-relator groups, Math. Proc. Camb. Phil. Soc. 109 (1991), 313–321. [28] O. Bogopolski, A surface analogue of a theorem of Magnus, Contemp. Math. 352 (2005), 55–89.

Baumslag, Fine, Rosenberger: One-relator groups

153

[29] O. Bogopolski and K. Sviridov, A Magnus theorem for some one-relator groups, in The Zieschang Gedenkschrift 14 (2008), 63–73. [30] A. K. Bousfield and D. M. Kan, Homotopy limits, completion and localization, Lecture Notes in Mathematics 304 (Springer-Verlag, 1972). [31] M. Bridson, and A. Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathemtischen Wissenschaften 319 (Springer-Verlag, Berlin, 1999). [32] K. Brown and E. Dror, The Artin-Rees property and homology, Israel J. Math. 22 (1975), 93–109. [33] A. M. Brunner, R. G. Burns and D. Solitar, The subgroup separability of free products with cyclic amalgamation, Contemp. Math. 33 (1984), 90–115. [34] I. Bumagin, O. Kharlampovich and A. Myasnikov, The isomorphism problem for fully residually free groups, J. Pure Appl. Alg. 20 (2007), 961–977. [35] J. O. Button and R. P. Kropholler, Non-hyperbolic free-by-cyclic and one-relator groups, ArXiv: 1503.01989, v1. [36] T. Camps, V. grosse Rebel and G. Rosenberger, Einf¨ uhrung in die kombinatorische und die geometrische Gruppentheorie (Heldermann-Verlag, 2008). [37] B. Chandler and W. Magnus, The History of Combinatorial Group Theory: A Case Study in the History of Ideas (Springer-Verlag, New York, 1982). [38] L. Ciobanu, B. Fine and G. Rosenberger, The surface group conjecture: cyclically pinched and conjugacy pinched one-relator groups, Results in Mathematics 64 (2013), 175–184. [39] M. Cohen and M. Lustig, Very small actions on R-trees and Dehn twist automorphisms, Topology 34 (1985), 575–617. [40] P. M. Curran, Subgroups of finite index in certain classes of finitely presented groups, J. Algebra 122 (1989), 118–129. [41] F. Dahmani and V. Guiradel, The isomorphism problem for all hyperbolic groups, Geometric and Functional Analysis 21 (2011), 223–300. ¨ [42] M. Dehn, Uber unendliche diskontinuierliche Gruppen, Math. Ann. 71 (1912), 116– 144. [43] J. L. Dyer, Separating conjugates in free products with amalgamation and HNN extension, J. Austral. Math. Soc. 29 (1980), 35–51. [44] J. Edmonds, Maximum matching and a polyhedron with 0, 1-vertices, J. Res. Nat. Bur. Standards Sect. B 69B (1965), 125–130. [45] D. B. A. Epstein, J. Cannon, D. Holt, S. Levy, M. Paterson and W. Thurston, Word Processing in Groups (James and Bartlett Publishers, Boston, London, 1992). [46] M. Feighn and M. Handel, Mapping tori of free group automorphisms are coherent, Ann. Math. 149 (1999), 1061–1077. [47] B. Fine, A. Gaglione, A. Myasnikov, G. Rosenberger and D. Spellman, A classification of fully residually free groups, J. Algebra 200 (1998), 571–605. [48] B. Fine, A. Gaglione, A. Myasnikov, G. Rosenberger and D. Spellman, The Elementary Theory of Groups (De Gruyter, 2016). [49] B. Fine, A. Gaglione, G. Rosenberger and D. Spellman, Something for Nothing: Some consequences of the solution to the Tarski problems, Proc. Groups St Andrews, 2013 (Cambridge University Press, 2015). [50] B. Fine, A. Gaglione, G. Rosenberger and D. Spellman, Elementary free groups, Cont. Math. 633 (2015), 41–58. [51] B. Fine, J. Howie and G. Rosenberger, One-relator quotients and free products of cyclics, Proc. Amer. Math. Soc. 102 (1988), 1–6. [52] B. Fine, O. Kharlampovich, A. Myasnikov, V. Remeslennikov and G. Rosenberger, On the surface group conjecture, Scient. Series A (2007). [53] B. Fine, O. Kharlampovich, A. Myasnikov and V. Remeslennikov, Tame automor-

154

Baumslag, Fine, Rosenberger: One-relator groups

phisms of elementary free groups, Comm. Algebra 42 (2014), 3386–3394. [54] B. Fine, M. Kreuzer and G. Rosenberger, Faithful real representations of cyclically pinched one-relator groups, Internat. J. Group Theory 3 (1) (2014), 1–8. [55] B. Fine, F. Levin and G. Rosenberger, Subgroups and decompositions of one-relator products of cyclics: Part 2: Normal torsion-free subgroups, J. Indian Math. Soc. 49 (1989), 237–247. [56] B. Fine, A. Moldenhauer and G. Rosenberger, Faithful real representations of groups of F-type, to appear. [57] B. Fine, A. Myasnikov, V. grosse Rebel and G. Rosenberger, A classification of CSA, commutative transitive and restricted Gromov one-relator groups, Result. Math. 50, 2007, 183–193. Erratum, Result in Math. 61 (2012), 421–422. [58] B. Fine and A. Peluso, Amalgam decompositions of one-relator groups, J. Pure Appl. Algebra 141 (1999), 1–11. [59] B. Fine, F. R¨ohl and G. Rosenberger, On HNN groups whose three generator subgroups are free, Inf. Groups and Group Rings, World Scientific 1 (1993), 13–36. [60] B. Fine, A. Rosenberger and G. Rosenberger, A note on Lyndon properties in onerelator Groups, Results Math. 59 (2012), 239–250. [61] B. Fine and G. Rosenberger, Generalizing algebraic properties of Fuchsian groups, Groups St Andrews 1989 – London Math. Soc. Lecture Notes Series 159 (1990), 124–148. [62] B. Fine and G. Rosenberger, On Restricted Gromov Groups, Comm. Algebra 20 (8) (1992), 2171–2182. [63] B. Fine and G. Rosenberger, The Freiheitssatz of Magnus and its extensions, Cont. Math. 169 (1994), 213–252. [64] B. Fine and G. Rosenberger, Algebraic generalizations of discrete groups (Marcel Dekker, 2000). [65] B. Fine and G. Rosenberger, Surface groups within Baumslag doubles, Proc. Edinburgh Math. Soc. 54 (1) (2011), 91–97. [66] B. Fine, G. Rosenberger and M. Stille, The isomorphism problem for a class of parafree groups, Proc. Edinburgh Math. Soc. 40 (1997), 541–549. [67] B. Fine, G. Rosenberger and M. Stille, Conjugacy pinched and cyclically pinched one-relator groups, Revista Math. Madrid 10 (1997), 207–227. [68] J. Fisher, A. Karrass and D. Solitar, On one-relator groups having elements of finite order, Proc. Amer. Math. Soc. 33 (1972), 297–301. [69] R. Fricke and F. Klein, Vorlesungen u ¨ber die Theorie der automorphen Funktionen, vols. 1, (1897) and 2 (1912), Teubner Leipzig (Reprinted: Johnson, New York, 1965). [70] A. Gaglione, S. Lipschutz and D. Spellman, Almost locally free groups and a theorem of Magnus, J. Groups, Complexity and Cryptology 1 (2009), 181–198. [71] A.M. Gaglione and D. Spellman, Even more model theory of free groups, in Infinite Groups and Group Rings (World Scientific Press, 1993), 37–40. [72] G. Gardam and D.J. Woodhouse, The geometry of one-relator groups satisfying a polynomial isoperimatric inequality, ArXiv: 1711,08155v2. [73] D. Gildenhuys, O. Kharlampovich and A. Myasnikov, CSA groups and separated free constructions, Bull. Austral. Math. Soc. 52 (1995), 63–84. [74] C. Gordon and H. Wilton, Surface subgroups of doubles of free groups, ArXiv 0902.3693v1. [75] M. Gromov, Hyperbolic groups, in Essays in Group Theory (MSRI Publications, Springer-Verlag, 1987), 75–263. [76] V. Guba, Equivalence of infinite systems of equations in free groups and semigroups to finite subsystems, Mat. Zametki 40 (1986), 321–324. [77] M. Gutierrez, Homology and completions of groups, J. Algebra 51 (1978), 354–366.

Baumslag, Fine, Rosenberger: One-relator groups

155

[78] J. Howie, On the asphericity of ribbon disc complements, Trans. Amer. Math. Soc. 280 (1985), 281–302. [79] S. Ivanov and P. Schupp, On the hyperbolicity of small cancellation groups and one-relator groups, Trans. Amer. Math. Soc. 350 (1998), 1851–1894. [80] A. Juhasz, Small cancellation theory with a unified small cancellation condition, J. London Math. Soc. 4 (1989), 57–80. [81] A. Juhasz, Solution of the conjugacy problem in one-relator groups, in Algorithms and classification in combinatorial group theory, (G. Baumslag and C. F. Miller (eds.)), MSRI (1992), 60–81. [82] A. Juhasz, Some applications of small cancellation theory to one-relator groups and one-relator products. in Geometric group theory, Vol.1, London Math. Soc. Lecture Notes Ser. 181 (Cambridge U. Press, Cambridge, 1993), 132–137. [83] A. Juhasz and G. Rosenberger, On the combinatorial curvature of groups of F-type and other one-relator products of cyclics, Cont. Math. 169 (1994), 373–384. [84] R.H. Haring-Smith, Groups and simple languages, Trans. Amer. Math. Soc. 279 (1983), 337–356. [85] I. Kapovich and P. Schupp, Genericity, the Arzhantseva-Olshanskii method and the isomorphism problem for one-relator groups, Math. Ann. 331 (2005), 1–19. [86] A. Karrass, W. Magnus and D. Solitar, Elements of finite order in a group with a single defining relation, Comm. Pure Appl. Math. 13 (1960), 57–66. [87] A. Karrass and D. Solitar, The subgroups of a free product of two groups with an amalgamated subgroup, Trans. Amer. Math. Soc. 150 (1970), 214–224. [88] O. Kharlampovich and A. Myasnikov, Irreducible affine varieties over a free group: I. Irreducibility of quadratic equations and Nullstellensatz, J. Algebra 200 (1998), 472–516. [89] O. Kharlampovich and A. Myasnikov, Irreducible affine varieties over a free group: II. Systems in triangular quasi-quadratic form and a description of residually free groups, J. Algebra 200 (1998), 517–569. [90] O. Kharlampovich and A. Myasnikov, Hyperbolic groups and free constructions, Trans. Amer. Math. Soc. 350 (2) (1998), 571–613. [91] O. Kharlampovich and A. Myasnikov, Description of fully residually free groups and irreducible affine varieties over free groups, Summer school in Group Theory in Banff, 1996, CRM Proceedings and Lecture notes 17 (1999), 71-81. [92] O. Kharlampovich and A. Myasnikov, Implicit function theorem for free groups, J. Algebra 290 (2005), 1–203. [93] O. Kharlampovich and A. Myasnikov, Effective JSJ decompostions, Contemp. Math. 378 (2005), 87–212. [94] O. Kharlampovich and A. Myasnikov, Elementary theory for free nonabelian groups, J. Algebra 302 (2006), 451–552. [95] O. Kharlampovich and A. Myasnikov, Algebraic Geometry over Free Groups, to appear. [96] O. Kharlampovich, A. Myasnikov, V. Remeslennikov and D. Serbin, Subgroups of fully residually free groups: algorithmic problems, Contemp. Math. 360 (2004). [97] S. H. Kim and S. I. Oum, Hyperbolic surface subgroups of one-ended doubles of free groups, J. Topology 7 (4) (2014), 927–947. [98] S. Kim and H. Wilton, Surface subgroups of doubles of free groups, preprint. [99] M. Lazard, Sur les groupes nilpotents et les anneaux de Lie, Annales Sci. L’Ecole Normale Superieure 3 (71) (1954), 101–190. [100] I. Leary, G. Niblo and D. Wise, Some free-by-cyclic groups, Groups St Andrews 1997 in Bath, II, London Math. Soc. Lecture Note Ser. 261 (Cambridge Univ. Press, Cambridge, 1999), 512–516.

156

Baumslag, Fine, Rosenberger: One-relator groups

[101] S. Lipschutz, The conjugacy problem and cyclic amalgamation, Bull. Amer. Math. Soc. 81 (1975), 114–116. [102] S. Liriano, The Nonisomorphism of Two One-Relator Groups, Ph.D. Thesis CUNY, (1993). [103] K. I. Lossov, SQ-universality of free products with amalgamated finite subgroups, Sibirsk. Mat. Zh. 27 (6) (1986), 128–139. [104] L. Louder and H. Wilton, Stackings and the W-cycles conjectures, Can. Math. Bull. 60 (3) (2017), 604–612. [105] L. Louder and H. Wilton, One-relator Groups with Torsion are Coherent, preprint. [106] A. Lubotzky, A group theoretic characterization of linear groups, J. Algebra 113 (1988), 207–214. [107] R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory (Springer-Verlag, 1977). ¨ [108] W. Magnus, Uber diskontinuierliche Gruppen mit einer definierden Relation (Der Freheitssatz), J. Reine Angew. Math. 163 (1930), 141–165. [109] W. Magnus, A. Karrass and D. Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations (Interscience Publishers, New York, London, Sydney, 1966). [110] A. I. Malcev, On faithful representations of infinite groups by matrices, Amer. Math. Soc. Translations 45 (1965), 1–18. [111] N. Mendelsohn and R. Ree, Free subgroups of groups with a single defining relation, Arch. Math. 19 (1968), 577–580. [112] Y. I. Merzlyakov, ed., Kourovka Notebook – Unsolved Problems in Group Theory (1980). [113] D. I. Moldavanskii, Certain subgroups of groups with one defining relation, Sibirsk. Math. Zh. 8 (1967), 1370–1384. [114] A. G. Myasnikov and V. N. Remeslennikov, Exponential groups 1: Foundations of the theory and tensor completion, Siberian Mat. J. 5 (1994), 1106–1118. [115] B. B. Newman, Some results on one-relator groups, Bull. Amer. Math. Soc. 74 (1968), 568–571. [116] G. Ponzoni, On the surface group conjecture (Thesis, Univ. Milan-Bicocca, 2014). [117] S. J. Pride, The isomorphism for two generator one-relator groups with torsion is solvable, Trans. Amer. Math. Soc. 227 (1977), 109–139. [118] E. S. Rapaport, Proof of a conjecture of Papakyriakopoulos, Ann. Math. 2 (1964), 608–613. [119] V. N. Remeslennikov, ∃-free groups, Siberian Mat. J. 30 (1989), 998–1001. [120] E. Rips and Z. Sela, Cyclic splittings of finitely presented groups and the canonical JSJ decomposition, Ann. Math. 146 (1) (1997), 53–109. [121] G. Rosenberger, Zum Isomorphieproblem f¨ ur Gruppen mit einer definierenden Relation, Ill. J. Math. 20 (1976), 614–621. [122] G. Rosenberger, Gleichungen in freien Produkten mit Amalgam, Math. Z. 173 (1980), 1–12: Berichtigung: Math. Z. 118 (1981), 579. [123] G. Rosenberger, Eine Bemerkung zu einer Arbeit von R. C. Lyndon, Archiv. der Math. 40 (1983), 200–207. [124] G. Rosenberger, The isomorphism problem for cyclically pinched one-relator groups, J. Pure Appl. Algebra 95 (1994), 75–86. [125] G. Sacerdote and P. Schupp, SQ-universality in HNN groups and one-relator groups, J. London Math. Soc. 7 (1974), 733–740. [126] Z. Sela, The isomorphism problem for hyperbolic groups I, Ann. Math. 141 (1995), 217–283. [127] Z. Sela, Diophantine geometry over groups I: Makanin-Razborov diagrams, Publ.

Baumslag, Fine, Rosenberger: One-relator groups

157

Math. de IHES 93 (2001), 31–105. [128] Z. Sela, Diophantine geometry over groups II: Completions, closures and formal solutions, Israel J. Math. 104 (2003), 173–254. [129] Z. Sela, Diophantine geometry over groups IV: An iterative procedure for validation of a sentence, Israel J. Math. 143 (2004), 17–130. [130] Z. Sela, Diophantine geometry over groups III: Rigid and solid solutions, Israel J. Math. 147 (2005), 1–73. [131] Z. Sela, Diophantine geometry over groups V: Quantifier elimination, Israel J. Math. 150 (2005), 1–97. [132] Z. Sela, Diophantine geometry over groups VI: The elementary theory of a free group, GAFA 18 (2006), 707–730. [133] A. Selberg, On discontinuous groups in higher dimensional symmetric spaces, Int. Colloq. Function Theory (Tata Institute, Bombay, 1960), 147–164. [134] P. Shalen, Linear representations of certain amalgamated products, J. Pure Appl. Algebra 15 (1979), 187–197. [135] V. Shpilrain, Recognizing automorphisms of the free groups, Arch. Math. 62 (1994), 385–392. [136] J. Stallings, Homology and central series of groups, J. Algebra 2 (1965), 170–181. [137] E. C. Turner, Test words for automorphisms of free groups, Bull. London Math. Soc. 28 (1996), 255–263. [138] B. A. F. Wehfritz, Generalized free products of linear groups, Proc. London Math. Soc. 27 (3) (1973), 402–424. [139] H. Wilton, One ended subgroups of graphs of free groups with cyclic edge groups, Geom. Topol. 16 (2012), 665–683. [140] D. Wise, Residual finiteness of positive one-relator groups, Comment. Math. Helv. 76 (2001), 314–338. [141] D. Wise, The residual finiteness of quasi positive one-relator groups with torsion, J. Algebra 66 (2002), 334–350. [142] D. Wise, The coherence of one-relator groups with torsion, and the Hanna Neumann Conjecture, Bull. London Math. Soc. 37 (5) (2005), 697–705. [143] D. Wise, The structure of groups with a quasiconvex hierarchy, Electronic Research Announcement in the Mathematical Sciences, AIMS 16 (2009), 44–55. [144] H. Zieschang, Automorphismen ebener discontinuierlicher Gruppen, Math. Ann. 166 (1966), 148–167. [145] H. Zieschang, E. Vogt and H.-D. Coldewey, Surfaces and planar discontinuous groups, Lecture Notes in Math. 835 (1980).

NEW PROGRESS IN PRODUCTS OF CONJUGACY CLASSES IN FINITE GROUPS ´ ∗ , MAR´IA JOSE ´ FELIPE† and CARMEN MELCHOR∗ ANTONIO BELTRAN ∗

Departamento de Matem´aticas, Universidad Jaume I, Av. Vicent Sos Baynat, s/n, 12071, Castell´on, Spain Email: [email protected], [email protected]

† Instituto Universitario de Matem´ atica Pura y Aplicada, Universidad Polit´ecnica de Valencia, Cam´ı de Vera, s/n, 46022, Valencia, Spain Email: [email protected]

Abstract Products of conjugacy classes is a well-established theme in Group Theory with open conjectures. We summarize known and new results concerning the influence that the product of two conjugacy classes exerts on the structure of a finite group. We add several open questions in order to inspire the reader to solve them and develop new techniques of research.

1

Introduction

Let G be a finite group. We will denote by xG the conjugacy class of an element x ∈ G. A subset X of G is said to be G-invariant if X g = {xg | x ∈ X} = X for all g ∈ G, and in this case, X is always union of conjugacy classes. We set η(X) to denote the number of distinct conjugacy classes appearing in X (observe that η(G) is the number of conjugacy classes of G). In particular, the product of two conjugacy classes A and B of G, that is, AB = {ab | a ∈ A, b ∈ B}, is a G-invariant set. There exist many results about the structure of a finite group regarding η(AB), some of which are related to the solvability and non-simplicity of the group. We introduce classic and current results about the normal structure of G from η(AB), specially η(AB) = 1 and η(AB) = 2, and some particular cases such us η(A2 ) ≤ 2, or η(AA−1 ) ≤ 3, where A−1 is the inverse class of A. In [8], Z. Arad and M. Herzog present results on the covering number for nonabelian simple groups. A regular covering theorem guarantees that for a nonabelian simple group G there exists a positive integer m such that for every conjugacy class C = 1 of G, we have C m = G. The authors collect results about upper bounds to the minimal value of m, named the convering number of G. Moreover, the authors conjecture one of the most significant problems related to the product of two conjugacy classes, that is, if there exist two conjugacy classes A and B in a group G such that η(AB) = 1, then G is not a non-abelian simple group [8, p.3]. This conjecture remains open although it has been verified for plenty of families of simple groups. The information of this survey is divided into the following three sections. First, in Section 2, we include notes in the framework of Arad and Herzog’s conjecture

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explaining the most recent advances on it. In Section 3, we present results about the square of conjugacy classes. We address the particular case in which the square of a class is a class and the simplest case when we square a real class. In Section 4, we discuss about the product of a conjugacy class by its inverse class. Along this overview, all groups considered are finite. Although we focus our attention on results concerning conjugacy classes, we will also make use of properties and characterizations appealing to characters.

2

Products of classes which are a conjugacy class

The first natural question about products of conjugacy classes is: When is the product of two classes a class again? It is trivial to see that if we take a central element z and a conjugacy class A of a finite group G, then zA is again a conjugacy class. A more general elementary fact is that if A and B are conjugacy classes such that their sizes are coprime numbers, then AB is a conjugacy class. In particular, if G is a nilpotent group and x and y are elements of G having coprime orders, then xG y G is the conjugacy class (xy)G . E. Adan-Bante obtained in [2] the following result concerning the case we are dealing with in this section by adding the hypothesis that the centralizers of the representatives of the classes are equal. Theorem 2.1 (Theorem A of [2]) Let G be a finite group, aG and bG be conjugacy classes of G. Assume that CG (a) = CG (b). Then aG bG = (ab)G if and only if [ab, G] = [a, G] = [b, G] and [ab, G] is a normal subgroup of G. Another relevant result was given in [14] by E. C. Dade and M. K. Yadav. They classified all those finite groups in which the product of any two non-inverse conjugacy classes is always a conjugacy class. A group G is said to satisfy Hypothesis A if for every x, y ∈ G such that xG = (y −1 )G , the equality xG y G = (xy)G holds. Theorem 2.2 (Theorem A of [14]) The group G satisfies Hypothesis A if and only if G is isomorphic to exactly one of the following: 1. a finite abelian group; 2. a non-abelian Camina p-group; 3. the semidirect product of the additive group F + and the multiplicative group F ∗ of some finite field F ; 4. the semidirect product of the elementary abelian group of order 9 and the quaternion group. We recall that a finite p-group G is said to be a Camina p-group if every nontrivial coset of G = [G, G] is a single conjugacy class of G. A group G is said to satisfy Hypothesis B if for every x, y ∈ G such that xG Z(G) = (y −1 )G Z(G), the equality xG y G = (xy)G holds. Also, groups satisfying Hypothesis B were classified.

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Theorem 2.3 (Theorem B of [14]) A finite group G satisfies Hypothesis B if and only if it is isoclinic to a group satisfying Hypothesis A, and thus to one of the list of Theorem 2.2. With regard to Arad and Herzog’s conjecture, the following result of Arad and E. Fisman (Theorem 2.4) is a consequence of the proof of Szep’s conjecture (Theorem 1 of [15]) which asserts that if a finite group G is the product of two subgroups with non-trivial centers, then G is not a non-abelian simple group. Hence, Arad and Herzog’s conjecture is solved for the following particular case. Theorem 2.4 (Theorem 2 of [15]) Let G be a finite non-abelian simple group, and let C and D be any pair of non-trivial conjugacy classes in G. Then (|C|, |D|) = 1. As it has been said in the Introduction, Arad and Herzog’s conjecture is still open, although it was checked by themselves in [8] for alternating groups, Suzuki groups, PSL2 (q), non-abelian simple groups of order less than one million, and 15 of the 26 sporadic simple groups. More recently, by using a characterization of the fact that the product of two conjugacy classes is a conjugacy class in terms of characters, J. Moori and H. P. Tong-Viet proved in [22] the conjecture for the families of simple groups listed in Theorem 2.6. Actually, the following lemma can also be used to check it for all the sporadic simple groups (for instance by using the Atlas [13] or GAP [16]). Lemma 2.5 (Preliminaries of [22]) Let G be a group and let a, b ∈ G be nontrivial elements of G. The following conditions are equivalent: 1. aG bG = cG 2. χ(a)χ(b) = χ(c)χ(1) for all χ ∈ Irr(G). Theorem 2.6 (Theorems 2, 3, 4, and 5 of [22]) Consider the following list of simple groups: PSL3 (q), PSU3 (q), where q is a power of a prime p,2 G2 (q) with q = 32m+1 , PSp4 (q) where q is a power of a prime p, PSp2n (3) with n ≥ 2, and PSUn (2), with (n, 3) = 1 and n ≥ 4. If G is one of the previous groups, then the product of any two non-trivial conjugacy classes of G cannot be a single conjugacy class. A few years later, R. Guralnick, G. Malle and P. H. Tiep proved Arad and Herzog’s conjecture in several more cases. Theorem 2.7 (Propositions 3.1, 3.2 and 3.3 of [18]) conjecture holds for the following simple groups:

Arad and Herzog’s

1. Un (q) with 3 ≤ n ≤ 6, (n, q) = (3, 2) 2. S4 (q), S6 (q), O8 (q) and O8 (q) 3. 2 B2 (22f +1 ) (f ≥ 1), 2 G2 (32f +1 ) (f ≥ 1), G2 (q) (q ≥ 3), 3 D4 (q), 2 F4 (22f +1 ) (f ≥ 1)

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In fact, the results in [18] are stronger. It is proved that if G is a finite group of Lie type and A and B are non-trivial conjugacy classes, corresponding to either two semisimple elements or two unipotent elements, then AB is not a conjugacy class. In addition, by using some results for semisimple elements, the original proof of Szep’s conjecture is considerably shortened. Furthermore, although Arad and Fisman already proved in [15] that in Sn and An (the symmetric and alternating groups, respectively) the product of two classes is not a class (see also another proof by Adan-Bante and H. Verrill in Theorem A of [5]), the authors give a very short proof of a slightly stronger result (Theorem 1.4 of [18]). Taking in mind that there is an openly accepted analogy between conjugacy classes and irreducible characters of a group G, it is natural to wonder whether Arad and Herzog’s conjecture is true when it is transferred into irreducible characters. We realize that the answer is no. For instance, it is proved (Theorem 10 of [23]) that there exist two non-trivial irreducible characters of An , χ and ψ, such that χψ is an irreducible character if and only if n is a perfect square.

3

Square of a conjugacy class

One of the earliest results considering squares of conjugacy classes was provided by D. Chillag and A. Mann in [12]. Although the techniques are relatively elementary, the proofs are based on the solvability of the groups of odd order due to FeitThompson. Theorem 3.1 (Proposition 5.2 of [12]) Let G be a finite group in which the square of each conjugacy class is the union of at most two conjugacy classes. Then G is solvable. On the other hand, a particular case of Arad and Herzog’s conjecture was recently given by G. Navarro and Guralnick in [19] by considering the square of a conjugacy class. They proved that when a conjugacy class K of a finite group G satisfies that K 2 is again a conjugacy class, then G is not simple, although the result goes much further. Theorem 3.2 (Theorem A of [19]) Let G be a finite group, let x ∈ G, and let K = xG be the conjugacy class of x in G. Then the following are equivalent: 1. K 2 is a conjugacy class of G. 2. χ(x) = 0 or |χ(x)| = χ(1) for every χ ∈ Irr(G), and CG (x) = CG (x2 ). 3. K = x[x, G] and CG (x) = CG (x2 ). In this case, [x, G] is solvable. Furthermore, if x has order a power of a prime p, then [x, G] has a normal p-complement. The most relevant fact is that the authors demonstrated the solvability of the normal subgroup [x, G] (which coincides with K ) by means of the Classification of the Finite Simple Groups (CFSG). The key of this proof is that all elements of x[x, G] are G-conjugate.

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We explain now the authors’ contribution to this topic. Whereas in Theorem 3.1 it is assumed that all the squares of conjugacy classes are union of at most two conjugacy classes, a new question arises: What can be said about the group structure when the square of one conjugacy class is union of exactly two conjugacy classes, one of them being the trivial class? The answer appears in Theorem 3.3. The case in which the square of a conjugacy class is union of two arbitrary classes still remains open. Recall that a conjugacy class K of a group G is said to be real if K = K −1 . In terms of characters, this means that every irreducible character of G takes real values on K. Inspired by Theorem 3.2, we observe that if K is a non-trivial real conjugacy class of G, it trivially follows that K 2 can never be a conjugacy class unless K consists of a single central involution of G. In [10], we study the case in which a finite group G contains a real conjugacy class K such that η(K 2 ) = 2, which is the simplest case when considering real classes. Theorem 3.3 (Theorem A of [10]) Let K = xG be a conjugacy class of a finite group G and suppose that K 2 = 1 ∪ D, where D is a conjugacy class of G. Then D = [x, G] is either cyclic or p-group for some prime p, and so K = x [x, G] is solvable. More precisely, 1. Suppose that |K| = 2.

(a) If o(x) = 2, then K ∼ = Z2 × Z2 and Z2 ∼ = D ⊆ Z(G). ∼ Zn and D is cyclic. (b) If o(x) = n > 2, then K =

2. Suppose that |K| ≥ 3. (a) If o(x) = 2, then either K and D are 2-elementary abelian groups or D is a p-group and |K| = pr with p an odd prime and r a positive integer. (b) If o(x) > 2, then D is a p-elementary abelian group for some odd prime p. Furthermore, either o(x) = p or o(x) = 2p. In every case, |K /D | ≤ 2. The techniques for proving Theorem 3.3 are quite elementary although we make use of Glauberman’s Z∗ theorem [17] and a result of Y. Berkovich and L. Kazarin (Lemma 3 of [11]). Both require tools from modular representation theory, so Theorem 3.3 is based on it as well. Other two main ingredients of the proof are the classic Burnside’s classification of finite 2-groups having exactly one involution and the classification of groups of order 16. We remark, however, that we do not appeal to the CFSG. Example 3.4 We show several examples of each case of Theorem 3.3. In some of them, the SmallGroups library of GAP is used. The m-th group of order n in this library is identified by n#m. Case 1.a Let G = D2n = a, b | an = b2 = 1, b−1 ab = a−1 with n an even integer and K = bG , which satisfies K 2 = 1 ∪ D where D = (an/2 )G .

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Case 1.b We consider again G = D2n with n ≥ 3 and K = aG . Then K 2 = 1∪D where D = (a2 )G . Let us remark that if n is odd, then D = K whereas if n is even, |K /D | = 2. Case 2.a Let N = x1 × · · · × xr = Z2 × · · · × Z2 and let G = N Sr the wreath product of N and Sr . Then K = {x1 , · · · , xr } is a class of G such that K 2 = 1∪D where D = {xi xj |i = j} is a class, because Sr acts transitively on D, and o(xi xj ) = 2 for every i = j and |K| = r. Furthermore, D is 2-elementary abelian and |K /D | = 2. On the other hand, let G = 216#8 ∼ = ((Z3 × Z3 )  Z3 )  Q8 generated by two elements a, b of order 4 and an element c of order 3. The class K = (a2 )G satisfies that K 2 = 1 ∪ D where D = cG . Moreover, o(a2 ) = 2, o(c) = 3, |K| = 9 and |D| = 24. In addition, D is a non-abelian 3-group of order 27. Case 2.b Let a ∼ = Z5 and let b ∼ = Z8 acting on a by ab = a2 . Let G = a b and take K = (b4 a)G . We have K 2 = 1 ∪ D where D = aG , o(b4 a) = 10, o(a) = 5, |K| = 4 and |D| = 4. In this case D ∼ = Z5 and K ∼ = Z10 . We get another example if we take G = 72#41 ∼ = (Z3 × Z3 )  Q8 and K = sG , with s being an element of order 3. This class satisfies that K 2 = 1 ∪ K with |K| = 8. Likewise, K ∼ = Z3 × Z3 . As an application of Theorem 3.3, the following corollaries are obtained. The first one is a direct consequence of it. Corollary 3.5 (Corollary B of [10]) Let G be a finite group such that every non-central conjugacy class K satisfies that K 2 = 1 ∪ D, where D is a conjugacy class of G. Then G/F(G) is an elementary abelian 2-group. In order to prove the next corollary, we use Theorem 3.2, so the CFSG is needed. Corollary 3.6 (Corollary C of [10]) Let K be a conjugacy class of a finite group G such that K 2 is union of conjugacy classes, all of which are central except at most one. Then K is solvable. Suppose now that every conjugacy class K of a group G satisfies that K 2 is a conjugacy class. It is trivial that every real element must lie in Z(G) and must have order 2. In [12], Chillag and Mann described the groups in which every real element is a central element. Particularly, in Remark 5.5 of [12], the authors also claim, with omitted proof, that any group satisfying the first assumption is nilpotent. In [10], the authors included an extension of this result, whose proof is easy and only requires the part of Theorem 3.2 that does not need the CFSG. Corollary 3.7 (Corollary E of [10]) Let G be a finite group and let π be a set of primes. Suppose that K 2 is a conjugacy class for all conjugacy class K of πelements of G. Then G/Oπ (G) is nilpotent. In particular, if π = π(G), then G is nilpotent. The above extension is needed in order to determine the structure of those groups in which every conjugacy class satisfies that its square is union of conjugacy classes all of which are central except at most one.

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Corollary 3.8 (Corollary F of [10]) Let G be a finite group such that every conjugacy class K satisfies that K 2 is union of conjugacy classes, all of them being central except at most one. Let M/F(G) = O2 (G/F(G)). Then G/M is nilpotent and, consequently, G is solvable with Fitting length at most 3.

4

Multiplying a conjugacy class by its inverse class

The early results concerning the product of a conjugacy class by its inverse were obtained for supersolvable groups, nilpotent groups and p-groups by Adan-Bante in [1] and [4]. These are the following: Theorem 4.1 (Theorem A of [4]) For any finite supersolvable group G and any conjugacy class K of G, we have that dl(G/CG (K)) ≤ 2η(KK −1 ) − 1. And as a consequence, the following corollary was obtained: Corollary 4.2 (Corollary B of [4]) For any supersolvable group G and any conjugacy classes A and B of G such that AB ∩ Z(G) = ∅, we have that dl(G/CG (A)) ≤ 2η(AB) − 1. Theorem 4.3 (Theorem A of [1]) Let G be a finite p-group and x ∈ G and K = xG . Assume that |K| = pn . Then the product KK −1 is the union of at least n(p − 1) + 1 distinct conjugacy classes of G, i.e., η(KK −1 ) ≥ n(p − 1) + 1. Theorem 4.4 (Theorem B of [1]) Let n be a positive integer. Then there exists a finite set Tn of positive integers such that for any nilpotent group G and any conjugacy class K of G with η(KK −1 ) ≤ n, we have that |K| ∈ Tn . Theorem 4.5 (Theorem C of [1]) Let p be a prime number. Let G be a finite p-group and K a conjugacy class of G. Then one of the following holds: 1. |K| = 1 and η(KK −1 ) = 1. 2. |K| = p and η(KK −1 ) = p. 3. |K| ≥ p2 and η(KK −1 ) ≥ 2p − 1. In view of the above results, we wonder what happens in general when η(KK −1 ) = 2 or η(KK −1 ) = 3 for any group G. Before explaining our results, we point out that Arad and Fisman obtained in [6], by working on the complex group algebra C[G], the non-simplicity of a group under the assumption that the product of two conjugacy classes is exactly the union of these two classes or their inverses. Theorem 4.6 (Theorem A of [6]) If C and D are non-trivial conjugacy classes of a finite group G such that either CD = mC + nD or CD = mC −1 + nD, where m and n are non-negative integers, then G is not a non-abelian simple group.

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In the above theorem, the case C = D−1 cannot certainly occur but it inspired us to tackle the simplest case appearing when we multiply a class by its inverse. In [10], we prove that if η(KK −1 ) = 2, then G is not simple. We conjecture that K (and KK −1 ) is solvable in this situation. Nevertheless, we have only been able to prove it in some particular cases that we will see later. On the other hand, the fact that η(KK −1 ) = 3 does not imply that K or KK −1 is solvable. Actually, KK −1 may be even simple. For instance, if G = Sn for any n ≥ 5 and K is the conjugacy class of transpositions, then η(KK −1 ) = 3 and KK −1 = An . We study the particular case in which KK −1 = 1 ∪ D ∪ D−1 with D a conjugacy class of G, and we demonstrate that G cannot be simple by means of the CFSG and a character theoretical property characterizing such condition. Also in this case, the solvability of K remains an open question. Theorem 4.7 (Theorem A of [9]) Let K be a non-trivial conjugacy class of a finite group G and suppose that KK −1 = 1 ∪ D ∪ D−1 , where D is a conjugacy class of G. Then G is not a non-abelian simple group. In particular, this theorem holds if KK −1 = 1 ∪ D. We note that under the assumption of Theorem 4.7, if K is real, then it is easy to see that D is real as well. Thus, K 2 = 1 ∪ D and we recover the hypothesis of Theorem 3.3, which provides the solvability of K without employing the CFSG. To prove Theorem 4.7, we use the following characterization, in terms of characters, of the property appearing in that theorem. Theorem 4.8 (Theorem B of [9]) Let G be a group and x, d ∈ G. Let K = xG and D = dG . The following are equivalent: 1. KK −1 = 1 ∪ D ∪ D−1 2. For every χ ∈ Irr(G) |K||χ(x)|2 = χ(1)2 +

(|K| − 1) χ(1)(χ(d) + χ(d−1 )). 2

In particular, if D = D−1 , then KK −1 = 1 ∪ D if and only if for every χ ∈ Irr(G) |K||χ(x)|2 = χ(1)2 + (|K| − 1)χ(1)χ(d). As we have asserted before, the proof of Theorem 4.7 needs the CFSG. We consider the families of alternating groups, sporadic simple groups and simple groups of Lie type and we see for each of them that the theorem is true. Regarding the alternating groups, we have two possibilities. If the classes of an element x in Sn and in An coincide, then we apply a result due to Adan-Bante and Verrill, which asserts that in Sn (as it happens in An ) the product of two conjugacy classes A and B cannot be a conjugacy class, and give an explicit description of the minimum possible value for η(AB) (Theorem 4.9). On the contrary, if the classes of x in Sn and in An are different, we can use an easier result (Lemma 4.10). Nevertheless, there is an alternative proof in the original paper ([9]).

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Theorem 4.9 (Theorem A of [5]) Let Sn be the symmetric group of n-letters, n > 5 and α, β ∈ Sn \ 1. Then η(αSn β Sn ) ≥ 2 and, if η(αSn β Sn ) = 2 then either α or β is a fixed point free permutation. Assume that α is fixed point free. Then one of the following holds 1. n is even, α is the product of n/2 disjoint transpositions and β is either a transposition or a 3-cycle. 2. n is a multiple of 3, α is the product of n/3 disjoint 3-cycles and β is a transposition. Lemma 4.10 Let α, β ∈ An with n ≥ 6. Assume that αAn = αSn and β An = β Sn . Then η(αAn β An ) ≥ 5. For the sporadic simple groups, the Atlas [13] (or GAP [16]) allows to make a oneby-one analysis to show that there exists an irreducible character which does not satisfy Theorem 4.8(2). Finally, for the simple groups of Lie type we use properties of the Steinberg character to obtain a contradiction with Theorem 4.8(2) again. As we have said above, under the hypotheses of the particular case of Theorem 4.7, KK −1 = 1 ∪ D, the solvability of K remains unsolved. We have only been able to prove it in some specific cases (Theorem 4.11). Of course, the group G need not be solvable. The typical non-solvable situation is a group of type Z.S.2, where |Z| = 3 and Z is in the center of Z.S. In addition, S is an appropriated non-solvable group acted by an automorphism of order 2, such that the non-trivial elements of Z are conjugate by this automorphism. Theorem 4.11 (Theorem C of [9]) Let K be a conjugacy class of a finite group G and suppose that KK −1 = 1 ∪ D, where D is a conjugacy class of G. Then |D| divides |K|(|K| − 1) and K /D is cyclic. In addition, 1. If |D| = |K| − 1, then K is metabelian. More precisely, D is p-elementary abelian for some prime p. 2. If |D| = |K|, then K is solvable with derived length at most 3. 3. If |D| = |K|(|K| − 1), then K is abelian. The corresponding property for irreducible characters and for almost-quasi simple groups has already been considered by G. Malle. In [21], he classified the groups G with G/Z(G) almost-simple satisfying the analogous character-theoretical hypothesis. Perhaps it might be possible to classify the groups of this type satisfying the class condition of Theorem 4.11, so this is also an open problem. As we have indicated above, two examples could be 3.A6 .2 and 3.J3 .2. Now we address a more simple case of Theorem 4.11, when KK −1 = 1 ∪ K. Observe that K is necessarily real, so K 2 = 1 ∪ K, and it turns out that K is pelementary abelian because K is a minimal normal subgroup all whose non-trivial elements have the same order. Thus, we wonder what happens when KK −1 = 1 ∪ K ∪ K −1 and K = K −1 . The interesting point of this non-real case could be that perhaps more arithmetical information can be obtained. Following the parallelism between conjugacy classes and irreducible characters, we reflect on the problem of translating our results into Character Theory. As we

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have asserted before by means of an example, the fact that η(KK −1 ) = 3 does not imply the non-simplicity of the group. Something similar occurs when we work with irreducible characters. For example, if we consider the simple group PSL(2, 11), there exist three irreducible characters χ, ψ and ϕ such that χχ = 1 + ψ + ϕ (see for instance page 290 of [20]). When we try to transfer Theorem 4.7 into the framework of irreducible characters by taking the product of a character by its complex conjugate, we find in [3] that the author gives the structure of a finite solvable group G with χ ∈ Irr(G) such that χχ = 1G + m1 α1 + m2 α2 where α1 , α2 ∈ Irr(G) are non-principal characters and m1 and m2 are strictly positive integers. We are not aware, however, whether the above equality may hold in a simple group when α2 = α1 = α1 . Nevertheless, regarding the particular case of Theorem 4.7, if we take G = PSp2n (3), n ≥ 2, or G = PSUn (2), (n, 3) = 1, n ≥ 4, it is known that there exists a non-trivial character ψ ∈ Irr(G) such that ψψ = χ+1, with χ ∈ Irr(G) (see [21] for instance). Therefore, we conclude that simplicity may occur when the particular case of Theorem 4.7 is considered for characters. Acknowledgements The results in this paper are part of C. Melchor’s Ph.D. thesis, and she acknowledges the predoctoral grant PREDOC/2015/46, Universitat Jaume I. A. Beltr´ an and M.J. Felipe are supported by the Valencian Government, Proyecto PROMETEOII/2015/011. A. Beltr´ an and C. Melchor are also partially supported by Universitat Jaume I, grant P11B2015-77. References [1] E. Adan-Bante, Conjugacy classes and finite p-groups, Arch. Math. (Basel) 85 (2005), no. 4, 297–303. [2] E. Adan-Bante, Homogeneous products of conjugacy classes, Arch. Math. 86 (2006), 289–294. [3] E. Adan-Bante, Products of characters with few irreducible constituents, J. Algebra 311 (2007), 38–68. [4] E. Adan-Bante, Derived length and products of conjugacy classes, Israel J. Math. 168 (2008), 93–100. [5] E. Adan-Bante and H. Verrill, Symmetric groups and conjugacy classes, J. Group Theory 11 (2008), no. 3, 371–379. [6] Z. Arad and E. Fisman, An analogy between products of two conjugacy classes and products of two irreducible characters in finite groups, Proc. Edinburgh Math. Soc. 30 (1987), 7–22. [7] Z. Arad and E. Fisman, About products of irreducible characters and products of conjugacy classes in finite groups, J. Algebra 114 (1988), 466–476. [8] Z. Arad and M. Herzog, Products of conjugacy classes in groups, Lecture Notes in Mathematics, 1112, Springer-Verlag, Berlin, (1985). [9] A. Beltr´an, M.J. Felipe and C. Melchor, Multiplying a conjugacy class by its inverse in a finite group, Israel J. Math. 227 (2018), no. 2, 811–825. [10] A. Beltr´an, M.J. Felipe and C. Melchor, Squares of real conjugacy classes in finite groups, Ann. Mat. Pura Appl. 97 (2018), no. 2, 317–328.

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[11] Y. Berkovich and L. Kazarin, Indices of elements and normal structure of finite groups, J. Algebra 283 (2005), no. 2, 564–583 (2005). [12] D. Chillag and A. Mann, Nearly odd-order and nearly real finite groups, Comm. Algebra 26 (1998), no. 7, 2041–2064. [13] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups, Oxford University Press, Eynsham (1985). [14] E.C. Dade and M. K. Yadav, Finite groups with many product conjugacy classes, Israel J. Math. 154 (2006), 29–49. [15] E. Fisman and Z. Arad, A Proof of Szep’s conjecture on non simplicity of certain finite groups, J. Algebra 108 (1987), 340–354. [16] GAP Group, GAP-Groups, Algorithms and Programming, Vers. 4.7.7 (2015) (http://www.gap-system.org) [17] G. Glauberman, Central elements in core-free groups, J. Algebra 4 (1966), 403–420. [18] R. Guralnick, G. Malle and Pham Huu Tiep, Products of conjugacy classes in finite and algebraic simple groups, Adv. Math. 234 (2013), 618–652. [19] R. Guralnick and G. Navarro, Squaring a conjugacy class and cosets of normal subgroups, Proc. Amer. Math. Soc. 144 (2016), no. 5, 1939–1945. [20] I. M. Isaacs, Character theory of finite groups, Academic Press, Inc, New York, (1976). [21] G. Malle, Almost irreducible tensor squares, Comm. Algebra, 27 (1999), no. 3, 1033– 1051. [22] J. Moori and H. P. Tong-Viet, Products of conjugacy classes in simple groups, Quaest. Math. 34 (2011), no. 4, 433–439. [23] I. Zisser, Irreducible products of characters in An , Israel J. Math. 84 (1993), no. 1–2, 147–151.

ASPHERICAL RELATIVE PRESENTATIONS ALL OVER AGAIN WILLIAM A. BOGLEY∗ , MARTIN EDJVET† and GERALD WILLIAMS§ ∗

Department of Mathematics, Oregon State University, Kidder Hall 368, Corvallis, OR 97331-4605, USA Email: [email protected]

† School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, U.K. Email: [email protected] § Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester, Essex CO4 3SQ, U.K. Email: [email protected]

In gratitude to Stephen J. Pride for his friendship and mentorship over the years.

Abstract The concept of asphericity for relative group presentations was introduced twenty five years ago. Since then, the subject has advanced and detailed asphericity classifications have been obtained for various families of one-relator relative presentations. Through this work the definition of asphericity has evolved and new applications have emerged. In this article we bring together key results on relative asphericity, update them, and exhibit them under a single set of definitions and terminology. We describe consequences of asphericity and present techniques for proving asphericity and for proving non-asphericity. We give a detailed survey of results concerning one-relator relative presentations where the relator has free product length four.

1

Introduction

Consider a relative group presentation P = G, x | r . Thus, G is a group, x is a set disjoint from G, and r is a set of words in the free product G ∗ F , where F denotes the free group with basis x. The group defined by P is the quotient G(P) = (G ∗ F )/R where R is the normal closure of r in G ∗ F . Relative presentations and the groups they define have long been studied in connection with equations over groups (e.g., [30], [34, Theorem 3], [57, Section 6.2], [61]), where the objective is to determine conditions on P under which the natural homomorphism G → G(P) is injective, so that G can be viewed as a subgroup of G(P). When the relative presentation P is aspherical, in a suitable sense, the cohomology and finite subgroups of G(P) can be completely described in terms of those of G. In Section 2 we describe asphericity concepts in algebraic topological terms. Methods from combinatorial geometry for proving asphericity are the focus in Section 3. In Section 4 we survey asphericity classifications that have been

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obtained for one-relator relative presentations where the relator has free product length four.

2

Asphericity concepts

We refer the reader to [16], [17], [36], [69] for general aspects of algebraic topology and the cohomology of groups. A topological space X is aspherical if each spherical map S k → X, k ≥ 2, is homotopic to a constant map. A connected CW complex K ( is acyclic, in which case is aspherical if and only if its universal covering complex K ( K is actually contractible. The homotopy type of an aspherical CW complex K is uniquely determined by its fundamental group and if π1 K ∼ = G then K is called an Eilenberg-Mac Lane complex of type K(G, 1) and all homotopy invariants of the complex K are algebraic invariants of the group G. In particular, the augmented ( → Z provides a resolution of the trivial module Z by cellular chain complex C∗ (K) free ZG-modules, which in turn determines the homology and cohomology groups of G. Asphericity concepts for ordinary group presentations are well established [21], [56], [64]. For example, the Lyndon Identity Theorem [55] implies that each onerelator presentation for a group G is combinatorially aspherical, which means that a K(G, 1) can be constructed as in [24]. 2.1

Asphericity in terms of the cellular model

In his work on equations over locally indicable groups, Howie [38] employed topological methods involving a relative two-complex (L, K) associated to a relative presentation P = G, x | r . Here, we take K to be a K(G, 1)-complex. The)complex L is then obtained by attaching to the one-point union K ∨ x Sx1 ) two-cells 1 1 via cellular loops ϕ˙ r ): S → K ∨ x Sx that realize the homotopy classes [ϕ˙ r ] ≡ r ∈G∗F ∼ = π1 (K ∨ x Sx1 ): *  L=K∨ Sx1 ∪ c2r . x

r

The relative two-complex (L, K(G, 1)) is the cellular model of P. Because K = K(G, 1) is chosen to be aspherical, the homotopy types of L and of the pair (L, K) are both uniquely determined by the relative presentation P. We denote L by L(P). We are interested in the situation where L(P) is aspherical and so is a K(G(P), 1). The following technical lemma will be used repeatedly. Lemma 2.1 Suppose that (Y, X) is a relative two-complex where X and Y are connected CW complexes and the inclusion-induced homomorphism π1 X → π1 Y is injective. If π2 Y = 0, then Y is aspherical if and only if X is aspherical. Proof Let p : Y( → Y be the universal covering projection. Since π1 X → π1 Y is injective, each component of the pre-image p−1 (X) is a copy of the universal ( and so Hk (p−1 (X)) is a direct sum of copies of Hk X. ( Since covering complex X

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π2 Y ∼ = H2 Y( and the complement Y − X has no cells in dimensions three = π2 Y( ∼ and up, the result follows by considering the long exact homology sequence for the pair (Y( , p−1 (X)).  The term ‘aspherical’ has been applied to relative group presentations in a variety of ways. The most general of these is given in Definition 2.2 below. This concept appeared implicitly in [12, Theorem 4.1] and explicitly as a defined concept in [14]; it is expressed in terms of the second relative homotopy group π2 (L(P), K(G, 1)) of the cellular model. This formulation is broad enough to accommodate previous incarnations of the term ‘aspherical’, but is also specific enough to activate the principal group-theoretic and topological consequences of asphericity such as those described in Theorem 2.4 and Theorem 2.19, below. In addition to the intrinsic value of a flexible definition of a basic concept, this definition of asphericity clearly delineates the concept of non-asphericity in an algebraic way. This is reflected in recent and ongoing work, such as [10], [14], [15], [45], [58], in which, when studying certain families of relative presentations, interesting groups and presentations are uncovered in the non-aspherical realm. Definition 2.2 (See [12, Section 4], [14]) A relative presentation P = G, x | r is aspherical if the second relative homotopy group π2 (L(P), K(G, 1)) is trivial. Lemma 2.3 A relative presentation P = G, x | r is aspherical if and only if the natural homomorphism G → G(P) is injective and L(P) is a K(G(P), 1)-complex. Proof Letting (L, K) = (L(P), K(G, 1)), the natural homomorphism G → G(P) is identified with the inclusion-induced homomorphism π1 K → π1 L, which in turn appears in the long exact sequence for the pair (L, K): 0 = π2 K → π2 L → π2 (L, K) → π1 K → π1 L → 1. This shows that if π2 (L, K) = 0 then G → G(P) is injective and π2 L = 0, so that L is aspherical by Lemma 2.1. The converse statement follows immediately from the long exact sequence.  If P = G, x | r is aspherical, then not only is the natural homomorphism G → G(P) injective, so that we can view G as a subgroup of G(P), but G plays a determining role in the subgroup and homological structure of G(P). More precisely we have the following. Theorem 2.4 (Compare [12]) Let P = G, x | r be an aspherical relative presentation. (a) If Γ is a subgroup of G(P) such that Γ ∩ G = 1, then Γ has geometric dimension at most two. (b) For n ≥ 3, the embedding G → G(P) determines natural equivalences of functors defined on the category of ZG(P)-modules: G(P)

Hn (G, ResG

(−)) → Hn (G(P), −), G(P)

H (G(P), −) → H n (G, ResG n

(−)).

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(c) ([46, Section 3], attributed to Serre; see also [29] and [44, Theorem 5]) Every finite subgroup of G(P) is conjugate to a subgroup of G. Moveover if w ∈ G(P) and the intersection G ∩ wGw−1 contains a non-trivial element of finite order, then w ∈ G. Proof Let (L, K) = (L(P), K(G, 1)). Given Γ ≤ G(P) as in (a), there is a covering projection p : L → L for which π1 L ∼ = Γ. Since Γ ∩ G = 1, the pre-image p−1 (K) is 1). a disjoint union of copies of the contractible universal covering complex K(G, Since the inclusion of p−1 (K) in L is a cofibration, the quotient map q : L → X that collapses each component of p−1 (K) to a zero-cell is a homotopy equivalence, so X is an aspherical two-complex with fundamental group Γ. To prove (b), denote ( and K. ( Since G embeds in G(P) by the simply connected covering complexes by L ( ∼ ( in dimensions Lemma 2.3, there is a chain isomorphism C∗ (L) = ZG(P) ⊗G C∗ (K) ∗ ≥ 3, so if M is an arbitrary (right) ZG(P)-module and n ≥ 3, then1 ( Hn (G(P), M ) = Hn (M ⊗G(P) C∗ (L)) ∼ ( = Hn (M ⊗G(P) ZG(P) ⊗G C∗ (K)) ∼ Hn (M ⊗G C∗ (K)) ( = G(P) ∼ Hn (G, Res (M )). = G

Similar calculations involving Hom lead to cohomology isomorphisms as in (b). The statement (c) is a direct consequence of (b) as discussed in the cited references.  Remark 2.5 If G is trivial and K = K(G, 1) is a point, then P = G, x | r is an ordinary group presentation with cellular model L and asphericity of P is equivalent to (topological) asphericity of L (see, e.g., [69]). The conclusions of Theorem 2.4 are relativized versions of standard facts about aspherical two-complexes. In particular, the fundamental group of an aspherical two-complex is torsion-free and has geometric and cohomological dimension at most two. The asphericity status of a relative presentation P = G, x | r is unaffected if any relator r ∈ r is replaced by a conjugate wr w−1 where  = ±1 and w ∈ G∗F because the homotopy type of the cellular model is unaffected by such a change. Thus one can assume that the relators of P are cyclically reduced. However, asphericity does impose some combinatorial restrictions on the relator set r ⊆ G ∗ F . Lemma 2.6 Let P = G, x | r be an aspherical relative presentation. (a) No relator r ∈ r is conjugate in G ∗ F to an element of G. (b) If r, s ∈ r and s is conjugate to r in G ∗ F , then r = s and  = 1. (c) If r ∈ r, then r is not expressible as a proper power in G ∗ F : if r = ˚ re in G ∗ F , then e = ±1. 1 For the isomorphisms when n = 3, the fact that each three-cell of L is contained in K  → M ⊗G(P) C2 L)  = ker(M ⊗G C3 K  → M ⊗G C2 K)  and implies that ker(M ⊗G(P) C3 L  M ) → HomG(P) (C3 L,  M )) = im(HomG (C2 K,  M ) → HomG (C3 K,  M )). im(HomG(P) (C2 L,

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) Proof Let (L, K) = (L(P), K(G, 1)) and K1 = K ∨ x Sx1 ⊆ L. Since π2 L = 0, the long exact sequence for the pair (L, K1 ) shows that the homotopy boundary ∂ : π2 (L, K1 ) → π1 K1 is injective. For each r ∈ r, the two-cell c2r of L has a characteristic map ϕ2r : (B 2 , S 1 ) → (L, K1 ) with homotopy class [ϕr ] ∈ π2 (L, K1 ). The Hurewicz homomorphism h2 : π2 (L, K1 ) → H2 (L, K1 ) carries the corresponding homotopy classes [ϕr ] ∈ π2 (L, K1 ), r ∈ r, to a free Z-basis for the cellular homology group H2 (L, K1 ) ∼ = Zr . To prove (a), just suppose that r = wgw−1 ∈ r where g ∈ G and w ∈ G∗F . Using the homotopy action of π1 K1 ∼ = G ∗ F on π2 (L, K1 ), the element C = w−1 · [ϕr ] ∈ π2 (L, K1 ) satisfies ∂C = g ∈ G ∼ = π1 K, so C is in the image of the inclusion-induced homomorphism π2 (L, K) → π2 (L, K1 ). On the other hand h2 (C) = h2 ([ϕr ]) = 0, so this contradicts asphericity of P. For (b), if r = wsw−1 in G∗F , then the element Δ = [ϕr ]− (w·[ϕs ]) ∈ π2 (L, K1 ) satisfies ∂Δ = r− wsw−1 = 1 in π1 K1 ∼ = G ∗ F and so Δ so is trivial in π2 (L, K1 ). Applying h2 , we therefore have the Z-linear relation 0 = h2 (Δ) = −h2 ([ϕr ]) + h2 ([ϕs ]) involving members of a basis of H2 (L, K1 ) ∼ = Zr . This implies that  = 1 and r = s in r. For (c), suppose that r = ˚ re ∈ r ⊆ G ∗ F . We show that e = ±1. The element −1 Σ = (˚ r · [ϕr ])[ϕr ] ∈ π2 (L, K1 ) satisfies ∂Σ = ˚ rr˚ r−1 r−1 = 1 in π1 K1 ∼ = G ∗ F , so ( Σ = 1 in π2 (L, K1 ). Letting p : L → L be the simply connected covering space, ( p−1 (K1 )), a choice of basepoints determines an isomorphism π2 (L, K1 ) → π2 (L, ( p−1 (K1 )) → which we then compose with the Hurewicz homomorphism h2 : π2 (L, ( p−1 (K1 )). In this way, the elements [ϕr ] pass to a free ZG(P)-basis for H2 (L, + ( p−1 (K1 )) ∼ the equivariant cellular chain group H2 (L, = r ZG(P) consisting of 2 preferred lifts c˜r of the two-cells of L − K. Under this process, the element Σ ∈ ( p−1 (K1 )), so the fact that Σ is trivial in π2 (L, K1 ) π2 (L, K) lifts to (˚ r −1)˜ c2r ∈ H2 (L, implies that ˚ r − 1 = 0 in ZG(P), whence ˚ r = 1 in G(P). Thus there exists D ∈ π2 (L, K1 ) such that ∂(D) = ˚ r ∈ π1 K1 ∼ = G ∗ F , so that the element De [ϕr ]−1 is in the kernel of the homotopy boundary ∂. Again, the fact that π2 L = 0 implies that De [ϕr ]−1 is trivial in π2 (L, K1 ) and so 0 = h2 (De [ϕr ]−1 ) = eh2 (D) − h2 ([ϕr ]). Since h2 ([ϕr ]) is part of a Z-basis for H2 (L, K1 ), this implies that e = ±1.  Definition 2.7 (Orientability [12]) A relative presentation P = G, x | r is orientable if it satisfies the relator conditions in Lemma 2.6(a) and (b). Thus every aspherical relative presentation is orientable. The term ‘orientable’ is used because it implies that no relator is a cyclic permutation of its inverse; in particular, no relator can be written in the form r = wg1 w−1 g2 where w ∈ G ∗ F and g1 , g2 ∈ G both have order two. See [12, Section 1.1]. Note that orientability does not exclude the presence of proper power relators. See Remark 3.3(c). Remark 2.8 (Proper powers) In the presence of proper power relators r = ˚ re(r) ∈ r ⊆ G ∗ F , where e(r) is the (maximal) exponent for r, an ) expanded cellular model M (P) is obtained from the one-point union K(G, 1) ∨ x Sx1 by attaching a copy of a K(Ze(r) , 1)-complex in place of the two-cell c2r so that K(G, 1) ⊆ L(P) ⊆ M (P). The construction is described in detail in [12, Section 4] and is

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based upon one that was introduced by Dyer and Vasquez [24]. In [12, Theorem 4.3] it was shown that if π2 (M (P), K(G, 1)) is trivial, then M (P) is a K(G(P), 1)complex and that the root ˚ r of each relator generates a cyclic subgroup ˚ r of order e(r) in G(P). (See [46, Proposition 1] or [12, page 36].) Cohomology calculations and classifications of finite subgroups follow as in Theorem 2.4(b),(c), see [12, (0.3),(0.4)] for details. The property π2 (M (P), K(G, 1)) = 0 for the relative presentation P is analogous to the concept of combinatorial asphericity for ordinary presentations [21]. The long exact homotopy sequence for the triple (M (P), L(P), K(G, 1)) implies that the inclusion-induced homomorphism π2 (L(P), K(G, 1)) → π2 (M (P), K(G, 1)) is surjective since π2 (M (P), L(P)) is trivial by the Cellular Approximation Theorem [36, Theorem 4.8]. In this survey we focus on the stronger condition π2 (L(P), K(G, 1)) = 0 of Definition 2.2 in order to simplify the narrative and to emphasize relativization of topological asphericity for two-complexes, as in Theorem 2.4(a). 2.2

Weak asphericity

An alternative (weaker) asphericity concept that is not tied directly to the π1 embedding question was introduced in [6] and studied further in [23], [1], [50], [51]. For the cellular model (L, K) = (L(P), K(G, 1)) of a relative presentation P = G, x | r , we can assume that the two-skeleton K 2 is modeled on a presentation Q , for for G so that the two-skeleton L2 is modeled on a lifted ordinary presentation P G(P) [12, Section 1.6]. The second homotopy modules π2 K 2 , π2 L2 have sometimes , (e.g., [13]). been denoted by π2 Q, π2 P Definition 2.9 (See [6], [23]) The relative group presentation P is weakly as, is generated as a ZG(P)pherical if the second homotopy module π2 L2 = π2 P module by the image of the inclusion-induced homomorphism π2 K 2 → π2 L2 , or equivalently if the inclusion-induced homomorphism π2 (K 2 ∪ L1 ) → π2 L2 is surjective. We note that the term ‘weakly aspherical’ was used with a related but slightly different meaning in [12] – see Remark 3.3(a). Lemma 2.10 (See [12, Lemma 1.7], [41, Theorem 4.2]) Let P = G, x | r be a relative presentation and let N be the kernel of the natural homomorphism G → G(P). Then: (a) P is weakly aspherical if and only if π2 L(P) = 0. (b) If P is aspherical, then P is weakly aspherical. (c) L(P) is a K(G(P), 1)-complex if and only if P is weakly aspherical and Hk (N, Z) = 0 for k ≥ 3. (d) If G embeds in G(P), then P is weakly aspherical if and only if P is aspherical. Proof Letting (L, K) = (L(P), K(G, 1)), the statement (a) follows from a diagram

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chase in the following ladder of long exact homotopy sequences. π2 (K 2 ∪ L1 ) −−−→ π2 L2 −−−→ π2 (L2 , K 2 ∪ L1 ) −−−→ π1 (K 2 ∪ L1 ) −−−→ π1 L2 ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ∼ ∼ ∼ =. =. =. . 0

−−−→ π2 L −−−→ π2 (L, K ∪ L1 ) −−−→ π1 (K ∪ L1 ) −−−→ π1 L

That the inclusion-induced map π2 (L2 , K 2 ∪ L1 ) → π2 (L, K ∪ L1 ) is an isomorphism follows from Whitehead’s description of the relative group π2 (Y, X) as a free crossed π1 X-module when Y is obtained from X by attaching two-cells [69, Theorem 2.9],[71]. Note that the inclusion-induced homomorphism π2 L2 → π2 L is surjective by the Cellular Approximation Theorem [36, Theorem 4.8]. With Lemma 2.3, the statement (b) is now obvious. Looking more closely at the proof ( → L is the universal covering projection, then each compoof Lemma 2.1, if p : L nent of the pre-image p−1 (K) has fundamental group isomorphic to N and so is (∼ a K(N, 1)-complex because K is aspherical. Thus Hk L = Hk (p−1 (K)) is a direct sum of copies of Hk (N, Z) for k ≥ 3; the statements (c) and (d) follow at once.  Remark 2.11 The proofs of Lemma 2.6(b),(c) used only the fact that π2 L(P) = 0, so those results also apply to weakly aspherical relative presentations by Lemma 2.10(a). However, Lemma 2.6(a) does not hold for all weakly aspherical presentations. For example, if G = g ∼ = Z, then the relative presentation P = G, ∅ | g is weakly aspherical because L(P) is a disc. However P is non-aspherical because the natural homomorphism G → G(P) is not injective. Thus, asphericity and weak asphericity are distinct concepts. The next example also distinguishes asphericity from weak asphericity. Example 2.12 (Weakly aspherical but non-aspherical) Let G = g ∼ = Z and let P = G, x | gxg −1 x−2 , xgx−1 g −2 . Here G(P) is the trivial group and L(P) is a contractible two-complex. So P is weakly aspherical by Lemma 2.10(a). Since G → G(P) is not injective, P is non-aspherical by Lemma 2.3. Remark 2.13 An open question from the 1950’s asks if there is a one-relator relative presentation P = G, x | xe1 g1 . . . xel gl with G(P) = 1 and G = 1. This question has been attributed to Kervaire and to Laudenbach, see [56, Chapter I.6] and [57,  page 403]. Freedman [30] noted that in any such example the exponent sum li=1 ei is equal to ±1 and that the group G is perfect and has no proper (normal) subgroups of finite index (by a result of Gerstenhaber and Rothaus [34]). Klyachko proved that the group G must have torsion [52] and can be chosen to be simple [53]. Arguing as in Lemma 2.10(c), if we further assume that L(P) is contractible, then G is acyclic. 2.3

Preliminary reductions

If the natural homomorphism G → G(P) is injective, then the long exact homotopy sequence for the pair (L, K) = (L(P), K(G, 1)) shows that π2 L ∼ = π2 (L, K), which

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means that elements of the relative homotopy group are represented by spherical maps S 2 → L. The implicit involvement of both π1 and π2 in Definition 2.2 (i.e., Lemma 2.3) also means that we can reduce to the case where the coefficient group is generated by the occurrences of coefficients in the relative relators. Lemma 2.14 Given a relative presentation P = G, x | r , if H is any subgroup of G for which the set of relators r ⊆ G ∗ F is contained in the free product H ∗ F , then P is aspherical if and only if P0 = H, x | r is aspherical. Proof Since H ≤ G, we can construct a K(G, 1)-complex K that contains a K(H, 1)-complex K0 as a subcomplex and then the cellular models (L, K) and (L0 , K0 ) are such that L = K ∪K0 L0 . Since the inclusion of K0 in K induces the monomorphism H → G, it follows that if either P0 or P is aspherical, then the natural maps H → G(P0 ) and G → G(P) are both injective and there is an amalgamated free product decomposition G(P) = G ∗H G(P0 ); see, e.g., [39, page 89]. We may therefore assume that H → G(P0 ), G → G(P), and G(P0 ) → G(P) ( → L be the universal covering projection, the are all injective. Letting p : L ( = p−1 (K) ∪p−1 (K ) p−1 (L0 ) shows that Mayer-Vietoris sequence for the union L 0 ∼ π2 L0 and so the ∼ H2 L ∼ H2 (p−1 (L0 )) is a direct sum of copies of H2 L (= /0 = π2 L = result follows from Lemma 2.10.  For one-relator relative presentations P = G, x | r , many asphericity classifications can be reduced to case where the exponents occurring on x are relatively prime. Lemma 2.15 Suppose that |d| ≥ 2 and consider the relative presentations P = G, x | r and V = G, y | s where r = xde1 g1 . . . xde g and s = y e1 g1 . . . y e g . Then P is aspherical if and only if V is aspherical and the element y has infinite order in G(V).2 Proof There are homotopy equivalences L(P) = K(G, 1) ∨ Sx1 ∪ c2r  K(G, 1) ∨ Sx1 ∨ Sy1 ∪ c2r ∪ c2y=xd  K(G, 1) ∨ Sx1 ∨ Sy1 ∪ c2s ∪ c2y=xd = L(V) ∨ Sx1 ∪ c2y=xd showing that G(P) = π1 L(P) is obtained from G(V) = π1 L(V) by adjoining a dth root of y ∈ G(V) [61, Theorem 5.1]. In particular, G(P) ∼ = G(N ) where N is the one-relator relative presentation N = G(V), x | y = xd and so we have an amalgamated free product decomposition G(P) ∼ = G(N ) ∼ = G(V) ∗y=xd x . Thus we can assume that G embeds in both G(V) and in G(P). Suppose first that P is aspherical. It follows from Lemmas 2.1 and 2.3 that both V and N are aspherical. Further, since |d| ≥ 2, the element x is not conjugate in 2

See Example 2.17 and Question 2.18 below.

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G(P) ∼ = G(N ) to an element of G(V), so Theorem 2.4(c) implies that x has infinite order in G(N ) and so y has infinite order in G(V). Conversely, if V is aspherical and y has infinite order in G(V), then we can prove that N is aspherical using the weight test [12, Theorem 2.1] (see Theorems 3.4 and 3.10 below) applied to the star graph N st of N : apply the weight −1 to the edge labeled y and apply the weight 1 to all remaining edges in the star graph. Thus, L(P)  L(V) ∨ Sx1 ∪ c2y=xd = K(G(V), 1) ∨ Sx1 ∪ c2y=xd = L(N ) and so P is aspherical.



In the case of balanced relative presentations (i.e., relative presentations P = G, x | r with |x| = |r|) a simple instance of asphericity occurs when G = G(P). Lemma 2.16 Consider a balanced relative presentation P = G, x | r for a group G(P) where |x| = |r| < ∞. If the natural homomorphism G → G(P) is an isomorphism, then P is aspherical. Proof Let (L, K) be the cellular model of P where K = K(G, 1). Assuming that G → G(P) is an isomorphism, we show that L is aspherical, so π2 (L, K) ∼ = π2 L = 0 and so P is aspherical. ( → L be the universal covering projection. The pre-image p−1 (K) is Let p : L connected (since the inclusion-induced homomorphism G = π1 K → π1 L = G(P) is ( surjective) and simply connected (since π1 K → π1 L is injective), so p−1 (K) = K ( ( ( is contractible. We therefore have a homotopy equivalence L  L/K = X where X is obtained by identifying p−1 (K) to a zero-cell c0 . We show that X, and hence ( is contractible, so that L is aspherical. Note that X is simply connected and L, two-dimensional, so it suffices to show that H2 X = 0. ( The group of deck transformations Aut(p) ∼ = G(P) freely permutes the cells of L ( invariant. Thus G(P) acts freely and cellularly leaving the subcomplex p−1 (K) = K on X − c0 and so the cellular chain complex of X consists of ZG(P)-modules ∂

∂

C2 X →2 C1 X →1 C0 X that are free of rank n = |x| = |r| in dimensions one and two and where C0 X = Z has trivial G(P)-action. The boundary operator ∂1 is trivial because X has just a ( is simply connected implies that the boundary single zero-cell. The fact that X  L n n operator ∂2 : ZG(P) → ZG(P) is a surjective endomorphism of the free ZG(P)module of rank n, which therefore has a right inverse: ∂2 ◦ σ = 1. A theorem of Kaplansky [49], [60] then implies that σ ◦ ∂2 = 1 and so H2 X = ker ∂2 = 0.  In connection with Theorem 2.4(c), we note that non-obvious torsion can occur in the aspherical setting. Example 2.17 (Torsion and asphericity) The group defined by the relative presentation J = g | g 4 , x | x4 gx−3 g 2 (denoted J4 (1, −3) in [14]) is cyclic of order four and g = x in G(J ). Thus J is aspherical by Lemma 2.16, even though x is a finite order element.

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In light of this example we pose the following question. Question 2.18 If P = G, x | xe1 g1 · · · xe g is an aspherical one-relator relative presentation and x determines an element of finite order in G(P), does  it follow that G → G(P) is an isomorphism? Note that the exponent sum i=1 ei must be equal to ±1, for otherwise the element x lies outside the normal closure of the natural image of G in G(P) so x has infinite order by Theorem 2.4(c). 2.4

Three-manifolds and zero divisors

A three-manifold group is the fundamental group of a (PL) three-manifold, which need not be compact, closed, or orientable. A result of Ratcliffe [66] concerning rational Euler characteristics for groups [70] was exploited in [45] to establish a connection between relative asphericity and the three-manifold status for the group G(P). We make this connection explicit in the following result. Theorem 2.19 (Three-manifold criterion) Suppose that the relative presentation P = G, x | r is such that G, x, and r are finite and the natural homomorphism G → G(P) is split by a retraction ν : G(P) → G. Assume that G(P) is a virtual three-manifold group. Then: (a) If P is aspherical, then |r| ≤ |x|; and (b) If |r| = |x| (i.e., P is balanced), then P is aspherical ⇐⇒ G → G(P) is an isomorphism. Proof Assume that P is aspherical. With notation as in the proof of Theorem 2.4(c), p : L → L is a regular covering projection with Aut(p) ∼ = G and the 1) is contractible. Thus X = L/K(G, 1)  L  pre-image p−1 (K(G, 1)) = K(G,

K(Γ, 1) where Γ = ker ν. Asphericity of X implies that χ(X) = 1 − |G|(|x| − |r|) is an invariant of π1 X ∼ = π1 L ∼ = Γ, denoted χ(Γ) as in [70]. If G(P), and hence Γ, is a virtual three-manifold group, then [66, Theorem 2(i)] implies that χ(Γ) ≤ 1 so |r| ≤ |x|, as in (a). If |r| = |x|, as in (b), then χ(Γ) = 1, so [66, Theorem 2(ii)] implies that Γ = 1, in which case G → G(P) is an isomorphism.  Example 2.20 (Virtual three-manifold groups) (a) Let G = g | g n where n ≥ 6 is even and let P = G, x | x2 g 2 x−1 g −1 . It is well known (see, for example, [48, page 196]) that G(P) ∼ = F (2, n)  Zn where F (2, n) = x0 , . . . , xn−1 | xi xi+1 = xi+2 (0 ≤ i ≤ n − 1) is a Fibonacci group. By [37] we have that F (2, n) is an (infinite) three-manifold group, so G(P) is a virtual three-manifold group and G(P) ∼ = g | g n . Thus P is non-aspherical by Theorem 2.19. (b) Let G = g | g n where n ≥ 6 and let P = G, x | x2 gx−1 g . As in part (a), we have that G(P) ∼ = S(2, n)  Zn where S(2, n) = x0 , . . . , xn−1 | xi xi+2 = xi+1 (0 ≤ i ≤ n − 1) is a Sieradski group. By [68], [18] we have that S(2, n) is an (infinite) three-manifold group, so again we have that P is non-aspherical.

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Ivanov [47] discovered a direct connection between non-weak asphericity for a one-relator relative presentation P and the existence of zero divisors in the group ring ZG(P). The following generalization of Ivanov’s result is due to Leary [54]. Theorem 2.21 ([54]) Let P = G, x | xe1 g1 · · · xe g be a one-relator relative presentation. Assume that the kernel  of the natural homomorphism G → G(P) is acyclic and that the exponent sum i=1 ei is nonzero. If P is non-weakly aspherical (in the sense of Definition 2.9), then the group ring ZG(P) contains a non-trivial  zero divisor that is a Z-linear combination of at most i=1 |ei | elements of G(P). Ivanov conjectured that if G is the fundamental group of an aspherical twocomplex and the natural homomorphism G → G(P) is injective, then P is aspherical (in the sense of Definition 2.2) if and only if G(P) is torsion-free [47]. A counterexample would require a torsion-free group whose integral group ring has a non-trivial zero divisor. That no such group exists is a longstanding open conjecture that is widely attributed to Kaplansky. Kim [50], [51] has obtained results on the asphericity of certain one-relator relative presentations with torsion-free coefficient group.

3

Methods from combinatorial geometry

A combinatorial geometric form of asphericity for ordinary presentations and their two-dimensional cellular models was described by Sieradski [67, Section 4], who gave a practical ‘coloring test’ for detecting the property. Gersten [33] termed the property diagrammatic reducibility (DR); he developed a flexible ‘weight test’ generalizing the coloring test and applied the DR property to the study of equations over groups. For a relative presentation P = G, x | r , any element of the kernel of the natural homomorphism G → G(P) leads to a map of pairs (B 2 , S 1 ) → (L(P), K(G, 1)). In [39, Lemma 1], Howie applied topological methods to show that any such map of pairs gives rise to a ‘relative diagram’ that emerges as the geometric dual to naturally occurring combinatorial structure called a ‘picture’. In this section we examine combinatorial geometric conditions on pictures that are sufficient to guarantee asphericity of P in the sense of Definition 2.2. 3.1

Diagrammatic reducibility

A picture P is a finite collection of pairwise disjoint discs {D1 , . . . , Dm } in the interior of a disc D2 , together with a finite collection 0 of pairwise disjoint simple arcs {α1 , . . . , αn }0 embedded in the closure of D2 − m i=1 Di in such a way that each arc meets ∂D2 ∪ m i=1 Di transversely in its end points [12]. The discs of a picture P are also called vertices of P and for this reason vi or simply v is often used in place of Di . The boundary of P is the circle ∂D2 , denoted by ∂P. For 1 ≤ 0i ≤ m, the corners of Di are the closures of the connected components of ∂Di − nj=1 αj , where ∂Di is the boundary of D i . The regions 0 0n Δ of P are the closures of the m connected components of D2 − D ∪ i i=1 j=1 αj . An inner region of P is a

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simply connected region of P that does not meet ∂P. (If P is connected note that any region that does not meet the boundary is simply connected.) 0 0n The picture P is non-trivial if m ≥ 1, is connected if m i=1 Di ∪ j=1 αj is connected, and is spherical if it is non-trivial and if none of the arcs meets the boundary of P. The number of arcs or edges in ∂Δ is called the degree of the region Δ and is denoted by d(Δ). A region of degree k will be called a k-gonal region. Now consider the relative presentation G, x | r and suppose that the picture P is labelled in the following sense: each arc αj is equipped with a normal orientation, indicated by a short arrow meeting the arc transversely, and labelled by an element of x ∪ x−1 . Each corner of P is oriented clockwise (relative to the disc/vertex to which the corner is associated) and labelled by an element of G. (We remark here that the convention anti-clockwise has been adopted by some authors.) If κ is a corner of a disc Di of P, then W (κ) will be the word obtained by reading in a clockwise order the labels on the arcs and corners meeting ∂Di beginning with the label on the first arc we meet as we read the clockwise corner κ. If we cross an arc labelled x in the direction of its normal orientation, we read x, else we read x−1 . A picture over P is a picture P labelled in such a way the following are satisfied: 1. For each corner κ of P, W (κ) ∈ r∗ , the set of all cyclic permutations of r∪r−1 which begin with a member of x. 2. If g1 , . . . , gl is the sequence of corner labels encountered in an anticlockwise traversal of the boundary of an inner region Δ of P, then the product g1 g2 . . . gl = 1 in G. We say that g1 g2 . . . gl is the label of Δ. A dipole in a labelled picture P over P consists of corners κ and κ of P together with an arc joining the two corners such that κ and κ belong to the same region and such that if W (κ)= Sg where g ∈ G and S begins and ends with a member of x ∪ x−1 , then W (κ )= S −1 g −1 . For example if x = {x} and r = {x3 gx−1 h} then a dipole is given in Figure 1(a). Definition 3.1 ([12], [23]) A relative presentation P is called diagrammatically reducible if every connected spherical picture over P contains a dipole. A picture P is called reduced if it does not contain a dipole. Thus if P fails to be diagrammatically reducible, or is non-diagrammatically reducible, then there is a non-trivial reduced, connected, spherical picture over P. A connected spherical picture P is called strictly spherical if the product of the corner labels in the annular region, taken in a clockwise direction, is equal to the identity in G. Definition 3.2 A relative presentation P is called weakly diagrammatically reducible if every connected strictly spherical picture over P contains a dipole. Remark 3.3 (a) Our definition of (weak) diagrammatically reducibility for a relative presentation coincides with (weak) asphericity as defined in [12]. (b) It is shown in [12, Lemma 1.7] that if P is weakly diagrammatically reducible and the natural map of G into G(P) is injective then P is diagrammatically reducible.

Bogley, Edjvet, Williams: Aspherical Relative Presentations Again

1

g

g¯

h

¯ h

1

1

1

−

+ 1

1

(a)

181

g

1

h

1

g −

+

h

(b)

Figure 1. (a) dipole; (b) disc/vertex contraction. (c) Diagrammatically reducible relative presentations can have proper power relators. See Remark 2.8 and Theorem 3.4(a) below. For two-complexes X, including two-dimensional cellular models of ordinary group presentations, the DR property (in the sense of [67], [33]) implies topological asphericity (π2 X = 0). For relative presentations that are orientable in the sense of Definition 2.7, the relationship between diagrammatic reducibility and asphericity is a consequence of [12, Theorem 4.1]. Theorem 3.4 (See [12, Theorem 4.1]) Let P = G, x | r be an orientable relative presentation. (a) If P is diagrammatically reducible, then π2 (M (P), K(G, 1)) = 0, where M (P) is the expanded cellular model described in Remark 2.8, and so the natural homomorphism G → G(P) is injective. (b) If P is diagrammatically reducible (resp. weakly diagrammatically reducible) and has no proper power relators, then P is aspherical (resp. weakly aspherical). The orientability condition in Theorem 3.4 is necessary to ensure that dipoles represent homotopically trivial elements of π2 (L(P), K(G, 1)). See [12, Figure 1] for an example of a dipole in the non-orientable setting and see [10, Example 3.5] for additional discussion. Diagrammatic reducibility and asphericity are distinct concepts for both ordinary and relative group presentations. Example 3.5 (Aspherical but non-diagrammatically reducible) (a) If the group G = g, h | h2 = 1, g = h2 ∼ = Z2 and P = G, x | x2 gx−1 h , then the natural homomorphism G → G(P) is an isomorphism, so P is aspherical. However, a reduced spherical picture P over P is depicted in Figure 2, so P is nondiagrammatically reducible. Cyclically reducing, we obtain P  = G, x | xh , which is obviously diagrammatically reducible. Thus diagrammatic reducibility is not robust under cyclic reduction of relators. (b) Let G = 1 and P = G, x, y | xyx−1 y −2 , yxy −1 x−2 . Taking K = K(G, 1) to be a point, the complex L(P) is a contractible, non-collapsible two-complex,

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g¯

g

1 ¯ h

1

h

1 h g

¯ h g¯ 1

Figure 2. Reduced spherical picture. so the relative presentation P is aspherical by Lemma 2.16. On the other hand, the result [22, Theorem 2.4] of Corson and Trace implies that L(P) is not DR (in the sense of [67], [33]). This in turn implies that the relative presentation P is non-diagrammatically reducible in the sense of Definition 3.1. In the next section we will discuss combinatorial geometric methods for detecting diagrammatic reducibility, as this concept is significant as a sufficient condition for asphericity of relative presentations. The next result shows that the diagrammatic reducibility concept also has independent group-theoretic and combinatorial significance. Theorem 3.6 Let P = G, x | r be a finite orientable relative presentation with no proper power relators. Assume that P is diagrammatically reducible and that the nat natural homomorphism G → G(P) admits a left inverse in the category of groups. Then the following are equivalent: nat

(a) G → G(P) is an isomorphism; nat

(b) the image of G → G(P) has finite index in G(P); (c) the cellular model L(P) collapses to its subcomplex K(G, 1): L(P) % K(G, 1). Thus, if L(P) does not collapse to K(G, 1), then G has infinite index in G(P). Proof Only the statement (b) ⇒ (c) requires proof. The fact that P is orientable and diagrammatically reducible means that every nonempty spherical picture over P contains a dipole. Since there are no proper power relators Theorem 3.4(b) and Lemma 2.3 imply that the cellular model *  L = L(P) = K(G, 1) ∨ Sx1 ∪ c2r x nat

r

is aspherical. The fact that G → G(P) has a left inverse ν : G(P) → G implies that there is regular G-covering p : L → L for which the quotient map q : L → X that 1)  ∗ to a point is a homotopy equivalence. As in collapses p−1 (K(G, 1)) = K(G, [10, Theorem 2.3], this implies that the two-complex X is the cellular model of a Gsymmetric presentation for the kernel of ν. Arguing as in the proof of [35, Lemma 3.1], any spherical picture (or diagram) K for X gives rise to a relative picture L over P and the presence of a dipole in L implies that the ordinary picture K also has a dipole. Now the proper power hypothesis for P implies that no two-cell of X is attached by a proper power, so it follows that X is diagrammatically reducible (DR)

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in the ordinary sense, as described by Gersten [33, Definition 3.1]. In particular X is aspherical in the topological sense: π2 X = 0. Since G(P) ∼ = π1 X  G, there is a bijection π1 X ≡ G(P)/G and so the hypothesis (b) implies that π1 X is finite. The fact that π2 X = 0 thus implies that X is simply connected and so by the result [22, Theorem 2.4] of Corson and Trace, X is a collapsible two-complex. A sequence of elementary collapses that reduces X to a point determines a sequence of elementary collapses that reduces L(P) to its subcomplex K(G, 1).  Example 3.7 (Diagrammatically reducible versus finiteness) (a) The natural homomorphism associated to the relative presentation J =  g | g 4 , x | x4 gx−3 g 2 of Example 2.17 is an isomorphism of cyclic groups of order four, so J is aspherical by Lemma 2.16. However the cellular model L(J ) does not collapse to the subcomplex K(g | g 4 , 1), so the relative presentation J is aspherical, yet it is not diagrammatically reducible, by Theorem 3.6. This means that J admits nonempty reduced spherical pictures. To our knowledge, no such pictures have yet been constructed. (b) Theorem 3.6 implies that the diagrammatically reducible relative presentations P = G, x | x2 gx−1 h obtained in [27] define infinite groups G(P). (More generally, it implies that if the relative presentation P = G, x | xk+1 gx−k h (k ≥ 1) is diagrammatically reducible then G(P) is infinite.) As we now describe, it follows that the (almost complete) classification of the finite cyclically presented groups H(n, m) and Gn (m, k) of [35], [72] can be obtained without the need of appealing to [62], [73]. The groups H(n, m) = Gn (m, 1) and Gn (m, k) each have Zn extensions defined by relative presentations of the form P = g | g n | x2 g m−k x−1 g k . The articles [35], [72] use the (almost complete) classification of diagrammatically reducible relative presentations G, x | x2 gx−1 h (see Theorem 4.16) to obtain corresponding (almost complete) classifications of the topologically aspherical cyclic presentations. Since groups defined by topologically aspherical presentations are torsion-free, it follows that the corresponding cyclically presented groups are either trivial or infinite. The articles [62], [73] use techniques from algebraic number theory and circulant matrices to classify the perfect groups H(n, m) and Gn (m, k). It transpires that, in the cases where the cyclic presentation is shown to be aspherical (which is precisely when the corresponding relative presentation P is known to be diagrammatically reducible), the group it defines is not perfect, and hence is non-trivial, and so is infinite. Theorem 3.6 allows us to sidestep the classifications of the perfect groups by concluding that in the diagrammatically reducible cases the group G(P), and hence the groups H(n, m) and Gn (m, k), are infinite. A number of ‘asphericity’ classifications in the literature are actually classifications of diagrammatic reducibility [12], [27], [43], [3], [4], [2], [28]. Some of these papers implicitly rely on [6, Lemma 3], which is stated and proved only for the concept of (weak) asphericity. As a result there is a slight gap in the literature due to the distinction between asphericity and diagrammatic reducibility. The following enhanced version of [6, Lemma 3] closes this gap.

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Lemma 3.8 Let P = G, x | r be a one-relator relative presentation where the relator r is cyclically reduced in the free product G ∗ F and is not an element of G (but may be a proper power in G ∗ F ). If the G-coefficients occurring in r are all contained in a free subgroup of G, then P is diagrammatically reducible. Proof Let P be a connected spherical picture over P. Since the corner labels of the picture lie in a free subgroup of G, we can assume that G is free (similar , over to Lemma 2.14). As in [12, Section 1.6], the picture P lifts to a picture P , for G(P). Since the relator r ∈ G ∗ F is an ordinary one-relator presentation P , is staggered in the sense of [56, cyclically reduced, the one-relator presentation P , Chapter III.9]. It follows from [56, Proposition III.9.7] that the lifted picture P contains a dipole and moreover this dipole is supported by a pair of discs joined by an arc labeled by the generator x. (Note that [56, Chapter III.9] permits the identification of a preferred subset of generators to support the staggered structure.) , actually detects a dipole in P. This means that the dipole in P  Remark 3.9 The proof that we have given for Lemma 3.8 depends on [56, Propositions III.9.7], which is proved in [56] by an extremely complex multiple induction. An alternative approach to these results appears in [40, Section 4], where the inductive complexities are sublimated within a conceptually transparent tower argument that ultimately rests on a structure theorem for staggered two-complexes X satisfying H 1 X = 0 [40, Lemma 3] (see also [33, Lemma 5.3]). All of these methods, including those in [56, Chapter III.9], imply that any staggered two-complex with no proper power relators is DR in the sense of [67], [33]. Further innovations to the staggering concept were applied to equations over groups by Anshel [5] and subsequently adapted to asphericity questions in [9]. We close this section with a brief discussion of combinatorial geometric methods that have been used by other authors in related contexts. The article [63] by Olshanskii includes concise historical remarks concerning geometric methods such as the van Kampen lemma, small cancellation theory, and aspherical presentations, as well as a survey of existence results in group theory obtained by the author using related geometric methods. The book [64] also develops a comprehensive treatment of geometric methods, including the formulation of a very general concept of asphericity for graded presentations. In [65, Theorem C(1)], Prishchepov adapts Olshanskii’s graded concept to the setting of one-relator relative presentations in which the relator has free product length four; Prishchepov’s result remains true as stated if one interprets the term ‘aspherical’ in the sense of Definition 2.2. We state this result in Theorem 4.20. In [20], Chalk proves that certain cyclic presentations are ‘aspherical’; his proof actually shows that these (ordinary) presentations are DR in the sense of [67], [33]. Using [10, Theorem 4.1], these results imply that certain relative presentations of the form g | g n , x | xr g l x−1 g −1 are aspherical in the sense of Definition 2.2. Chalk also discusses the relationship between asphericity for ordinary and relative presentations [20, page 1520].

Bogley, Edjvet, Williams: Aspherical Relative Presentations Again g

1

h

|k| − 1 edges 1

1 x ¯

185

x (l + k) − 2 edges

g

h x ¯

1 l − 1 edges

x

1 1 (b)

(a)

Figure 3. Star graphs. 3.2

Star graph and the weight test

In this section we outline methods for showing that a given relative presentation P is diagrammatically reducible. The star graph P st of P is a graph whose vertex set is x ∪ x−1 and edge set is r∗ . For r ∈ r∗ , write r = Sg where g ∈ G and S begins and ends with a member of x ∪ x−1 . The initial and terminal functions are given as follows: ι(r) is the first symbol of S, and τ (r) is the inverse of the last symbol of S. The labelling function on the edges is defined by λ(r) = g −1 and is extended to paths in the usual way. For example let x = {x} and r = {xl gxk h}. If l > 0 and k > 0 the star graph is given by Figure 3(a) and if l > 0 and k < 0 then the star graph is given by Figure 3(b). A non-empty cyclically reduced cycle (closed path) in P st will be called admissible if it has trivial label in G. Each inner region of a reduced picture over P supports an admissible cycle in P st . A weight function θ is a real-valued function on the set of edges of P st that satisfies θ(Sg) = θ(S −1 g −1 ) where Sg = r ∈ r∗ . The weight of a closed cycle is the sum of the weights of the constituent edges. A weight function is weakly aspherical if the following conditions are satisfied: (I) Let r ∈ r∗ , with r = xε11 g1 . . . xεnn gn . Then n 

ε

i−1 (1 − θ(xεi i gi . . . xεnn gn xε11 g1 . . . xi−1 gi−1 )) ≥ 2.

i=1

(II) The weight of each admissible cycle in P st is at least 2. A weakly aspherical weight function on P st is aspherical if each cyclically reduced closed cycle in P st has non-negative weight. Theorem 3.10 ([12, Theorem 2.1], [32], [42, Theorem 2.2]) If the star graph P st admits a (weakly) aspherical weight function, then P is (weakly) diagrammatically reducible. 3.3

Curvature distribution

Another method that can be used to show that a given relative presentation P is diagrammatically reducible is curvature distribution (see, for example, [26]). Assume

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that P is a non-trivial reduced, connected, spherical picture over P. We proceed as follows. Contract ∂P to a point, which is then deleted. This way the regions Δ of the amended picture, also called P, are simply connected and tesselate the 2-sphere. The region obtained from the annular region is called the distinguished region Δ0 and its label may or may not equal the identity in G; whereas the label of every other region is equal to the identity. (We remark here that when drawing it is often customary to shrink each disc/vertex of P to a point – an example is shown in Figure 1(b). In what follows we shall simply use the term vertex.) The next step is to define an angle function, that is, a real-valued function on the set of corners of P, with the further condition that the sum of all the angles at any given vertex is equal to 2π. This way each vertex has zerocuvature; and if Δ is a k-gonal region then the curvature c(Δ) is equal to 2π − ki=1 (π − θi ) where the sum is taken over the corner angles θi (1 ≤ i ≤ k) of Δ. Therefore the total curvature c(P) of P is given by c(P) = Δ∈P c(Δ). Given this, it is a consequence of Euler’s formula, for example, that c(P) = 4π and so P must contain regions of positive curvature. For example, note that each corner c of P corresponds to an edge e in the star graph P st and so if θ is a weight function on P st , then one possible angle function ˆ be the dual of P with angles induced by assigns π · θ(e) to the corner c of P. Let P those of P. Then the condition (I) for the weight function θ in Section 3.2 enforces ˆ has non-positive curvature, while the condition (II) for θ that each region of P ˆ has non-positive curvature since P is assumed to be implies that each vertex of P reduced. This leads to a contradiction to Euler’s formula (see, e.g., [13, Sect. 1.1]), showing that the relative presentation P is diagrammatically reducible. Note that by assigning angles according to a weight function on the star graph, the angles are assigned in a uniform way that is not sensitive to local configurations in the given picture P. We now describe a method, first introduced in this context in [27], that ultimately results in a more flexible assignment of angles. If we believe that P is diagrammatically reducible and so no reduced spherical picture P can exist, then we try to obtain a contradiction by showing that the total curvature is less than 4π. The strategy is to show that the positive curvature that may exist in P can be sufficiently compensated by negative curvature. To this end we systematically locate each Δ = Δ0 satisfying c(Δ) > 0 and distribute c(Δ) to ˆ of Δ. For such regions Δ ˆ define c∗ (Δ) ˆ to equal c(Δ) ˆ plus all the near regions Δ ˆ ˆ positive curvature Δ receives minus all the curvature Δ distributes as a result of  ˆ The final this distribution scheme. It is clear that c(P) is at most Δ∈P c∗ (Δ). ˆ ≤ 0 for Δ ˆ = Δ0 ; and that c∗ (Δ ˆ 0 ) < 4π. steps are to show that c∗ (Δ) Let v be a vertex of P. A standard angle function is to assign the angle 0 to each corner of v which forms part of a region of degree two and assigns the angles 2π/d(v) to the remaining d(v) corners. Therefore if Δ is a k-gonal region (k > 2) with vertices v1 , . . . , vk such that d(vi ) = di (1 ≤ i ≤ k) then c(Δ) = c(d1 , . . . , dk ) = (2 − k)π + ki=1 2π/di ; or if d(Δ) = 2 then c(Δ) = 0. Suppose that d(v) > 2 for each vertex v. Since c(3, 3, 3, 3, 3, 3) = 0 we have, for example, that if c(Δ) > 0 then 3 ≤ d(Δ) ≤ 5. Since the label of each region corresponds to an admissible closed path in the star graph, if we consider Figure 3(a) then only degree four is

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possible and if both g and h are involved this forces the label to be (up to cyclic permutation and inversion) one of gh−1 11−1 , gh−1 g1−1 , hg −1 h1−1 or gh−1 gh−1 and then either g = h, h = g 2 , g = h2 or (gh−1 )2 = 1 in G. A similar analysis of Figure 3(b) yields g = h±1 , g = h±2 or h = g ±2 in G.

4

One relator relative presentations with length four relator

Asphericity of one-relator relative presentations has been considered in the articles [12], [6], [27], [43], [23], [3], [4], [1], [2], [65], [28]

(1)

and in [59], [51], [50]. Those listed at (1) contain classifications of aspherical onerelator relative presentations when the relator has free product length four; that is for relative presentations Q = G, x | xl gxk h for certain values of the parameters l > 0, k = 0 and where g, h are non-trivial elements of G. Specifically, they provide detailed asphericity studies for the relative presentations Q for the cases {l, k} = {2, 1}, {3, 1}, {4, 1}, {3, 2}, {n, 1} (n ≥ 2), {l, k} (l, k > 0), {2, −1}, {3, −1}, {n, −1} (n ≥ 4), l > 0, k < 0 and l > 2|k| or |k| > 2l. Our purpose in this section is to survey the results from these articles, bring them up to date (where appropriate) and present them under the definition of asphericity given in Definition 2.2. We express some of our results in terms of the parameter μ = 1/|g| + 1/|h| + 1/|gh−1 |. 4.1

The Platonic case

We introduce the following condition: (P ) μ > 1 and g = h. For many values of l, k it has been shown that, when (P ) holds, spherical pictures based on the Platonic solids can be constructed, and hence Q is nondiagrammatically reducible. Further, in some cases these have been used to prove that Q is non-weakly aspherical. Specifically, we have the following results. Suppose that (P ) holds. If {l, k} = {2, 1}, {3, 2}, {4, 1}, {n, 1} (n ≥ 2), {2, −1}, {3, −1}, or if l > 0, k > 0 then Q is non-diagrammatically reducible by [12, Theorem 3.4], [43, Lemmas 10 and 12], [3, Lemma 3.4], [27, Lemma 4.3], [2, Lemma 3.1], or [4, Lemma 3.5(ii)] respectively. If {l, k} = {3, 1} or {3, −1} then Q is non-weakly aspherical by [6, Lemma 7] or [1, pages 71–75], respectively. If l ≥ 2, k = −1 and |g|, |h| ≥ 3 and (P ) holds then Q is non-weakly aspherical by [23, Lemmas 3.15, 3.16, 3.17]. We expect that these results hold more generally and may be extended to prove non-asphericity. We therefore make the following conjecture: Conjecture 4.1 If Case (P ) holds then Q is non-aspherical.

188 4.2

Bogley, Edjvet, Williams: Aspherical Relative Presentations Again Euclidean curvature and short admissible closed path cases

It is convenient to introduce the (possibly overlapping) cases (P ), (Z), (M ), (J4 ), (J6 ), (K5 ), (K6+ ), (K6− ), (L6 )

(2)

where (P ) is defined in Section 4.1 and the remaining cases are defined as follows: (Z) g = h and |g| < ∞; (M ) g = h−1 and |g| < ∞; (J4 ) (g = h2 , |h| = 4) or (h = g 2 , |g| = 4); (J6 ) (|g| = 2, |h| = 3, [g, h] = 1) or (|g| = 3, |h| = 2, [h, g] = 1); (K5 ) (g = h2 , |h| = 5) or (h = g 2 , |g| = 5); (K6+ ) (g = h2 , |h| = 6) or (h = g 2 , |g| = 6); (K6− ) (g = h−2 , |h| = 6) or (h = g −2 , |g| = 6); (L6 ) (g = h3 , |h| = 6) or (h = g 3 , |g| = 6). In Section 3 we saw that, amongst others, the presence of relations g = h±1 or g = h±2 or h = g ±2

(3)

lead to short admissible closed paths in the star graph and so present obstacles in proving asphericity for Q. In [14] it was suggested that the Euclidean curvature condition μ = 1 may sometimes also present a barrier to asphericity. The full picture is not yet clear, but it seems that the condition (3) interacts with the condition μ = 1 to present a liminal space where some presentations Q are aspherical and others are non-aspherical. While we believe that the cases listed at (2) capture many situations where non-aspherical presentations may occur, they are not (at least for k < 0) the only ones, as we shall see in Examples 4.2 and 4.3. The cases listed at (2) satisfy either (3) or μ ≥ 1 or both, in Case (P ) the group g, h is a quotient of a spherical triangle group and in the remaining cases it is cyclic. Specifically, Cases (P ) and (Z) satisfy μ > 1 and Cases (Z), (M ), (J4 ), (K6+ ), (K6− ) and (K5 ) all satisfy (3). The case (K5 ) simultaneously satisfies it in two ways: g = h±2 and h = g ∓2 . Cases (J4 ), (J6 ), (K6− ), (L6 ) satisfy μ = 1 with ({|g|, |h|}, |gh−1 |) = ({2, 4}, 4), ({2, 3}, 6), ({3, 6}, 2), ({2, 6}, 3), respectively. The remaining values of ({|g|, |h|}, |gh−1 |) that yield μ = 1 are ({3, 3}, 3), ({4, 4}, 2). Examples 4.2(a),(b), below, show that Q may be non-aspherical when g, h is noncyclic and ({|g|, |h|}, |gh−1 |) = ({2, 3}, 6) or ({3, 3}, 3). We are not aware of any non-aspherical example in the case ({4, 4}, 2). Example 4.2 (Finite groups outside the cases (2)) (a) Let G be the group G = g, h | g 2 , h3 , (gh)2 (g −1 h−1 )2 ∼ = S3 ⊕ Z3 and let Q = G, x | x2 gx−1 h . Here −1 we have (|g|, |h|, |gh |) = (2, 3, 6), and so μ = 1. Using GAP [31] we have that

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G(Q) is a finite, solvable, group of order 27216 = 24 · 35 · 7 and derived length 6. By Theorem 2.4(c) the presentation Q is non-aspherical. (b) Let G = g, h | g 3 , h3 , ghg −1 h−1 ∼ = Z3 ⊕ Z3 and let Q = G, x | x2 gx−1 h . Here we have (|g|, |h|, |gh−1 |) = (3, 3, 3), and so μ = 1. We have that G(Q) is a finite, solvable, group of order 13608 = 23 · 35 · 7 and derived length 5 and hence Q is non-aspherical. (c) Let G = g | g 8 ∼ = Z8 and let Q = G, x | x2 g 2 x−1 g . As was essentially first observed in [7, Table 4], the group G(Q) is a finite, solvable, group of order 2361960 = 23 · 310 · 5, and derived length 4. Hence Q is non-aspherical. As we shall see in Corollary 4.17, this settles a previously open case of [27, Theorem 1.1]. We may also prove non-asphericity of Q by exhibiting an element of prohibited finite order. Example 4.3 (Infinite groups with torsion outside the cases (2)) (a) Let G = g | g n where n ≥ 9 is odd and let Q = G, x | x2 g 2 x−1 g −1 . As in Example 2.20 we have that G(Q) ∼ = F (2, n)  Zn where F (2, n) = x0 , . . . , xn−1 | xi xi+1 = xi+2 (0 ≤ i ≤ n − 1) is a Fibonacci group. Setting w = x0 x1 . . . xn−1 , we have that w2 = 1 in F (2, n) (see [48, page 83]), and by (the proof of) [7, Proposition 3.1] w is an element of order exactly two. Thus G(Q) contains an element of order 2 so by Theorem 2.4(c) we have that Q is non-aspherical. (b) Let G = g | g 9 and let Q = G, x | x2 g 2 x−1 g . Then we have that G(Q) ∼ = H(9, 3)  Z9 where H = H(9, 3) = x0 , . . . , x8 | xi xi+3 = xi+1 (0 ≤ i ≤ 8) is a Gilbert-Howie group which, by [19, Lemma 15], is infinite. Setting w = x0 x 4 x 8 x 3 x 7 x 2 x 6 x 1 x5 , the diagram [74, Figure 2] (which was derived from [27, Figure 4.1(i)]) can be used to show that w2 = 1 in H. Moreover the group H/w H is finite of order 215 · 7. Thus w has order exactly two in H so G(Q) contains an element of order 2 and hence Q is non-aspherical, by Theorem 2.4(c). (This order two element w was first obtained in the proof of [45, Lemma 4].) (c) Let G = g | g 7 and let Q = G, x | x2 g 2 x−1 g . Then we have that G(Q) ∼ = H(7, 3)  Z7 where H = H(7, 3) = x0 , . . . , x6 | xi xi+3 = xi+1 (0 ≤ i ≤ 6) is a Gilbert-Howie group which, by [35, Theorem 3.3] (due to Thomas), is infinite. (Alternatively use GAP to show that the second derived subgroup of H is free abelian of rank 8). Setting w = x0 x3 x6 x2 x5 x1 x4 , we have that [74, Figure 1] (see also [27, Figure 4.1(h)]) can be used to show that w2 = 1 in H. Moreover the group H/w H is finite of order 128. Thus w has order exactly 2 in H so G(Q) contains an element of order 2 and hence Q is non-aspherical by Theorem 2.4(c).

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In Cases (J4 ), (J6 ), Theorem A of [14] shows that the relative presentation Q is aspherical only when the natural homomorphism G → G(P) is an isomorphism. Theorem 4.4 ([14, Theorem A]) Suppose that Case (Jn ) holds (n = 4 or 6). Then the following are equivalent: (a) Q is aspherical; (b) |l + k| = 1 and either l ≡ 0 mod n or k ≡ 0 mod n; (c) The natural homomorphism G → G(P) is an isomorphism. We now prove the corresponding result for Cases (Z) and (M ): Theorem 4.5 If the Case (Z) or (M ) holds, then the following are equivalent: (a) Q is aspherical; (b) |l + k| = 1 and l = 0 or k = 0; (c) The natural homomorphism G → G(P) is an isomorphism. Proof That (c) ⇒ (a) follows from Lemma 2.16. We show (b) ⇐⇒ (c). We have G(P) ∼ = G ∗g G0 where G0 has an ordinary presentation G0 ∼ = g, x | g n , xl gxk g for some n ≥ 2 and where  = ±1. Thus G → G(P) is an isomorphism if and only if G0 = g ∼ = Zn . When  = −1, [15, Theorem A(c)] shows that G0 = g ⇐⇒ (b) holds. Likewise, when  = 1, [58, Theorem 3.2.2.2] shows that G0 = g ⇐⇒ (b) holds. (These two theorems classify finiteness of groups of the form G0 = g, x | g n , xl gxk g .) Thus (b) ⇐⇒ (c). We now prove that (a) ⇒ (c). Given that Q is aspherical, it suffices to prove that G0 = g . By Lemma 2.14, the relative presentation Q0 = g , x | xl gxk g is aspherical and so by Theorem 2.4(c), it suffices to prove that G0 = G(Q0 ) is finite. We show that l+k = 0. Supposing otherwise, if l = −k, then Lemma 2.6(a) implies that l = 0 (and g = 1). We have xl gx−l g = 1 in G(Q) so g − ∈ g ∩ xl g x−l and Theorem 2.4(c) implies that xl ∈ g and so xl has finite order in G0 , contradicting the fact that if l = −k, then the quotient group G0 /g , which is generated by x, is infinite cyclic. Now suppose that  = 1, l + k = 0, and Q is aspherical. In this case, the element xl−k is central in G0 ∼ = g, x | g n , xl gxk g by [58, Lemma 3.2.2.2] so asphericity of Q0 implies that xl−k ∈ g by Theorem 2.4(c). Now l − k = 0 because otherwise the relator xl gxk g = (xl g)2 is a proper power, which contradicts asphericity by Lemma 2.6(c). Thus the element x ∈ G0 has finite order, so Theorem 2.4(c) implies that x is conjugate in G0 to an element of g . Consider the central quotient Δ = G0 /xl−k ∼ = g, x | g n , xl−k , (xl g)2 n l−k ∼ = g, x, u | g , x , (xl g)2 , u = xl ∼ = g, x, u | g n , xl−k , u(l−k)/(l,k) , (ug)2 , u = xl

∼ = g, u | g n , u(l−k)/(l,k) , (ug)2 ∗u=xl x | xl−k .

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The fact that x is conjugate to an element of g implies that this amalgamated free product decomposition is degenerate, that is x = u = xl , and so it follows that (l, k) = 1. Thus Δ ∼ = g, x | g n , xl−k , (xg)2 is the ordinary triangle group. The five-term homology sequence for the central extension is H2 G0 → H2 Δ → xl−k → H1 G0 → H1 Δ → 0. ∼ g, x | g n , xl gxk g has nonzero Since l + k = 0, the 2 × 2 relation matrix for G0 = determinant and thus H2 G0 = 0. Since x has finite order in G0 , this implies that H2 Δ is finite, which in turn implies that the triangle group Δ is finite and so G0 is finite, as required. It remains to consider the case where  = −1, l + k = 0, and Q is aspherical. We claim that x has finite order in G0 . To see this, arguing as in the proof of [15, n n Lemma 14], the relations gxk g −1 = x−l and g n = 1 imply that xk −(−l) = 1, so x has finite order unless k n −(−l)n = 0. If k n −(−l)n = 0, then since k = −l, it follows that k = l = 0 (and n is even) and so we have gxl g −1 = x−l , whence g 2 commutes with xl and in particular g 2 ∈ g ∩ xl g x−l . We cannot have g 2 = 1, for otherwise our relator is a proper power xl gxl g −1 = (xl g)2 , contrary to Lemma 2.6(c), so Theorem 2.4(c) implies that xl ∈ g and so x has finite order. Arguing as in the case  = 1, the fact that x has finite order in G0 implies that (l, k) = 1, and so [15, Lemma 14] shows that the group G0 ∼ = g, x | g n , xl gxk g −1 is a semi-direct ∼ product G0 = x  g . Thus G0 is finite, as required.  The remaining cases from (2) are (K5 ), (K6+ ), (K6− ), (L6 ). These appear to be challenging cases and only partial information is known. We now survey the state of knowledge for the values of {l, k} considered in the studies listed at (1). In some articles non-diagrammatic reducibility was proven by explicitly constructing spheres though these, in themselves, do not prove non-asphericity. However if the corresponding group G(Q) can be shown to be finite of order greater than six (which can sometimes be proved using GAP) an application of Theorem 2.4(c) proves non-asphericity. Table 1 summarizes what is known for each case. When G(Q) is known to be finite of order greater than six the cell either contains an expression [M, N ], which means that G(Q) is the N ’th group of order M in the Small Groups library [8], or it contains a number, which means that G(Q) is finite of that order, but does not have an entry in the Small Groups library. In these cases Q is non-aspherical by Theorem 2.4(c). The group of order 24530688 = 28 · 34 · 7 · 132 shows that Q is nonaspherical in the previously unresolved Case (E2) of [6], and was studied in detail in [11, Lemma 9.7]. In the case (K6 ) with l = 2, k = −1 the group G(Q) is the infinite virtual three-manifold group from Example 2.20(a), so Q is non-aspherical by Theorem 2.19. A question mark in a cell indicates that it remains unknown if the presentation Q is aspherical; these cases are either isolated as unresolved exceptional cases or are the subject of questions on their asphericity statuses in the references given in the final column. We note that aspherical presentations Q exist in the case (K5 ), as we shall see in Theorem 4.21.

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{l, k} {2, 1} {3, 1} {4, 1} {3, 2} {n, 1} (n ≥ 5) l, k > 0, {l, k} not above {2, −1} {3, −1}

(K5 ) [165,1] [1100,26] [3775,3] [2525,3] ? ?

(K6+ ) [378,8] ? ? ? ? ?

(K6− ) [342,7] ? ? ? ? ?

(L6 ) [342,7] 24530688 ? ? ? ?

Reference [12] [6] [43] [43] [3] [4]

[55,1] [110,2]

[336,210] ?

Ex. 2.20(a) ?

[54,6] 9072

[27] [2]

Table 1. Resolved and unresolved Cases (K5 ), (K6+ ), (K6− ), (L6 ) in the studies (1). 4.3

Asphericity results

We now survey results from the articles listed at (1) concerning asphericity and diagrammatic reducibility for relative presentations Q = G, x | xl gxk h . In many cases we shall assume that none of (2) hold, those cases having been discussed in Sections 4.1 and 4.2. For example, the results discussed in those sections imply that if {l, k} = {2, 1} or {2, −1} and (2) holds, then Q is not aspherical. See Theorem 4.9 and Corollary 4.17 below. We first consider the cases where l = ±k. Theorem 4.6 ([4, Lemma 3.3]) If l = k ≥ 1 then Q is diagrammatically reducible if and only if g = h or |g −1 h| = ∞. Corollary 4.7 If l = k = 0, then Q is aspherical if and only if |g −1 h| = ∞. Proof If g = h then the relator xl gxk h is a proper power (xl g)2 so Q is nonaspherical, by Lemma 2.6(c). Thus we may assume g = h. If |g −1 h| = ∞ then Q is diagrammatically reducible and orientable, hence aspherical. Thus we may assume that |g −1 h| < ∞. The relator xl gxl h implies that (xl g)2 = (g −1 h)−1 so xl g has finite order in G(Q). If Q is aspherical then Theorem 2.4(c) implies that xl g is conjugate to an element of G, i.e., xl g = whw−1 for some h ∈ G and w ∈ G ∗ x . Thus xl gwh−1 w−1 has exponent sum l in x and is equal to the identity of G(Q). But all words that are trivial in G(Q) are products of conjugates of elements of G or of the relator and so their exponent sum is a multiple of l + k = 2l, a contradiction.  Theorem 4.8 If l = −k = 0, then Q is aspherical if and only if |g| = |h| = ∞. Proof If |g| = |h| = ∞ then Q is orientable and an easy application of the weight test shows that Q is diagrammatically reducible, hence aspherical. Conversely, if Q is aspherical, then 1 = h−1 = x−k gxk ∈ G ∩ x−k Gxk and 1 = g −1 = xk hx−k ∈ G∩xk Gx−k and so Theorem 2.4(c) implies that |g| = |h| = ∞ because the elements x±k both lie outside the normal closure of G in G(Q) and so neither of the elements x±k is conjugate in G(Q) to any element of G – see Theorem 2.4(c). 

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From now on we assume that l = ±k, in which case the relative presentation Q = G, x | xl gxk h is orientable with non-proper power relator. Thus if Q is diagrammatically reducible, then it is aspherical. If G is torsion-free then Q is diagrammatically reducible by [50, Lemma 2]. The difficulty in proving diagrammatic reducibility comes about when G has torsion. We shall use the preceding observation freely to deduce asphericity when a given result asserts diagrammatic reducibility. We start with the case {l, k} = {2, 1}. Theorem 4.9 ([12]) Let {l, k} = {2, 1}. Then Q is diagrammatically reducible and hence aspherical if and only if none of the conditions (2) hold and when |g| < ∞, then we have g = h2 and h = g 2 . For the case {l, k} = {3, 1} we introduce the following conditions from [6]: (BBP-E4) {|g|, |h|} = {2, 4} and gp{g, h} ∼ = Z2 ⊕ Z 4 ; (BBP-E5) {|g|, |h|} = {2, 5} and gp{g, h} ∼ = Z2 ⊕ Z 5 . Theorem 4.10 ([6, Theorem 4]) Let {l, k} = {3, 1} and suppose that none of the conditions (2) hold. Suppose further that neither (BBP-E4) nor (BBP-E5) hold. Then Q is weakly aspherical and hence aspherical. We now turn to the case {l, k} = {4, 1}. Theorem 4.11 ([43, Theorem 2]) Let {l, k} = {4, 1} and suppose that none of the conditions (2) hold. Then Q is diagrammatically reducible and hence aspherical. Generalizing Theorems 4.10 and 4.11 we have the following for the case {n, 1}: Theorem 4.12 ([3, Theorem 1.1]) Let {l, k} = {n, 1} where n ≥ 3 and suppose that none of the conditions (2) hold. Then Q is diagrammatically reducible and hence aspherical. Note that Theorem 4.12 shows that Q is diagrammatically reducible and aspherical in the previously unresolved cases (BBP-E4), (BBP-E5) of Theorem 4.10. We now consider cases {l, k} with l > 0, k > 0 that are not (necessarily) of the form {n, 1}. For the case {l, k} = {3, 2} we introduce the following condition, (which is extracted from the statement of [43, Theorem 1]).3 (HM-E) (g = h2 , 6 < |h| < ∞) or (h = g 2 , 6 < |g| < ∞). Theorem 4.13 ([43, Theorem 1]) Let {l, k} = {3, 2} and suppose that none of the conditions (2) hold. Suppose further that (HM-E) does not hold. Then Q is diagrammatically reducible and hence aspherical. 3 There are typos in the statements of Lemma 9 and of parts (3) and (4) of Theorem 1 of [43]. The proof of [43, Lemma 9] employs coset enumeration to check that the groups defined by h, x | hk , x3 h2 x2 h are finite for 2 ≤ k ≤ 5. As in [43, Question 2], the asphericity status of g, x | x3 gx2 g 2  is unresolved if 6 < |g| < ∞.

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To consider the general case l > 0, k > 0, l = k we introduce the following condition from [4]: (AEJ-E)

(i) g = h2 , 6 < |h| < ∞, l < k < 2l; or

(ii) h = g 2 , 6 < |g| < ∞, k < l < 2k; or (iii) h = g 2 , 6 < |g| < ∞, l < k < 2l; or (iv) g = h2 , 6 < |h| < ∞, k < l < 2k. Theorem 4.14 ([4, Theorem 1.1]) Let l > 0, k > 0, l = k, {l, k} =  {2, 1} and suppose none of the conditions (2) hold. Suppose further that (AEJ-E) does not hold. Then Q is diagrammatically reducible and hence aspherical. We do not expect there to be any non-aspherical presentations in the Case (AEJE): Conjecture 4.15 Let l > 0, k > 0, l = k. If (AEJ-E) holds then Q is diagrammatically reducible and hence aspherical. We now turn to the cases l > 0, k = −1, where a less complete picture is known. We start with the case l = 2, k = −1. For this we need to introduce the following conditions: (E-E1) (|g| = 9, |h| = 3, h = g 3 ) or (|h| = 9, |g| = 3, g = h3 ); (E-E2) (|g| = 9, |h| = 3, h = g −3 ) or (|h| = 9, |g| = 3, g = h−3 ); (E-E3) (|g| = 8, |h| = 4, h = g 2 ) or (|h| = 8, |g| = 4, g = h2 ). Theorem 4.16 ([27, Theorem 1.1]) Suppose l = 2, k = −1 and that none of the conditions (2) hold and that none of (E-E1),(E-E2),(E-E3) hold. Then Q is diagrammatically reducible if and only if none of the following holds: (i) |g| < ∞ and either g = h−2 or h = g −2 ; (ii) [g, h] = 1 and either |g| = 2 or |h| = 2; (iii) {|g|, |h|} = {2, 3} and (gh)2 (g −1 h−1 )2 = 1; (iv) |g| = |h| = 3 and [g, h] = 1; (v) |g| = |h| = 7 and either g = h2 or h = g 2 ; (vi) |g| = |h| = 9 and either g = h2 or h = g 2 . The following corollary shows that the non-diagrammatically reducible presentations identified in Theorem 4.16, and also those in Case (E-E3), are non-aspherical. Corollary 4.17 Suppose l = 2, k = −1 and that neither of (E-E1) nor (E-E2) holds. Then Q is aspherical if and only if none of the conditions (2) hold, none of Cases (i)–(vi) of Theorem 4.16 hold, and (E-E3) does not hold.

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Proof It suffices to show that Q is non-aspherical in Cases (i)–(vi) and (E-E3). By Lemma 2.14 we may assume that G = g, h . In Case (i) we may assume that h = g −2 and |h| = n. If n ≤ 5 or n = 7 then G(Q) is finite of order greater than n, so Q is non-aspherical by Theorem 2.4(c); if n ≥ 6 is even then G(Q) is non-aspherical by Example 2.20(a), and if n ≥ 9 is odd then G(Q) is non-aspherical by Example 4.3(a). In Cases (iii), (iv), (E-E3) the presentation Q is non-aspherical by Example 4.2, and in Cases (v), (vi) it is non-aspherical by Example 4.3. In Case (ii), taking h2 = 1, then (as in [27]) the relation (xgh)2 = g −1 (xgh)g can be shown to hold in G(Q), from which it follows that xgh is an element of order 2|g| − 1 in G(Q), and so Q is non-aspherical by Theorem 2.4(c).  We now turn to the case l = 3, k = −1. Theorems 1.1 and 1.2 of [2] consider asphericity of one-relator relative presentations G, x | xg1 xg2 xg3 x−1 g4 (see also [1]). By putting g1 = g2 = 1, g = g3 , h = g4 , this becomes the presentation Q with {l, k} = {3, −1}. With these restrictions, the exceptional Cases (E) and (E4) of [2] paper become: (AAE-E) g, h ∼ = Z2 ⊕ Z4 ; (AAE-E4) (|h| = 8, g = h4 ) or (|g| = 8, h = g 4 ). As a corollary of [2, Theorems 1.1,1.2] we then have: Theorem 4.18 ([2, Theorems 1.1,1.2]) Suppose l = 3, k = −1 and that none of the conditions (2) hold. Suppose further that (AAE-E) and (AAE-E4) do not hold. Then Q is diagrammatically reducible and hence aspherical. We now turn to the case {l, k} = {n, −1} where n ≥ 4. This is considered in [23] under the hypothesis that g 2 = 1, h2 = 1 and the exclusion of three exceptional families: (D-E1) (g = h2 , 3 < |h| < ∞) or (h = g 2 , 3 < |g| < ∞); (D-E2) (g = h−2 , 3 < |h| < ∞) or (h = g −2 , 3 < |g| < ∞); (D-E4) (g = h3 , |h| = 9) or (h = g 3 , |g| = 9). Theorem 4.19 ([23]) Suppose l ≥ 4, k = −1, that g 2 = 1, h2 = 1 in G, and that none of (Z), (M ), (P ) hold. Suppose further that none of (D-E1),(D-E2),(D-E4) hold. Then Q is diagrammatically reducible and hence aspherical. Note that the hypotheses of Theorem 4.19 imply that none of the cases given in (2) hold. The more general case l > 0, k < 0 was considered in [65] under stronger hypotheses on relations involving the elements g, h of G. The following theorem, as stated in [65], includes the hypothesis that the natural homomorphism G → G(Q) is injective. That hypothesis is redundant however, as (under the remaining hypotheses) injectivity follows from [25].

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Theorem 4.20 ([65, Theorem C(1)]) Suppose that l > 0, k < 0 and l > 2|k| or |k| > 2l, that Cases (Z), (M ) do not hold, and that one of the following holds: (i) |g| ≥ 6, |h| ≥ 3, h = g ±2 , h = g −3 , g = h−2 ; or (ii) |g| ≥ 3, |h| ≥ 6, g = h±2 , g = h−3 , h = g −2 ; or (iii) |g| ≥ 4, |h| ≥ 4, g = h−2 , h = g −2 . Then Q is aspherical. We close with the following asphericity result, which was used to complete the classification of the finite Fibonacci groups F (r, n). Its proof highlights the intricacy of the arguments that can be required when proving asphericity of presentations Q when (3) holds. Theorem 4.21 ([28, Theorem 1.2]) Let l ≥ 7, k = −1 and suppose that (K5 ) holds. Then Q is diagrammatically reducible and hence aspherical. References [1] Abdul Ghafur Bin Ahmad. The application of pictures to decision problems and relative presentations. PhD thesis, University of Glasgow, 1995. [2] Abd Ghafur Bin Ahmad, Muna A. Al-Mulla, and Martin Edjvet. Asphericity of length four relative group presentations. J. Algebra Appl., 16(4):1750076, 2017. (27 pages). [3] Suzana Aldwaik and Martin Edjvet. On the asphericity of a family of positive relative group presentations. Proc. Edinb. Math. Soc., 60:545–564, 2017. [4] Suzana Aldwaik, Martin Edjvet, and Arye Juhasz. Asphericity of positive free product length 4 relative group presentations. Forum Math., to appear. [5] Iris Lee Anshel. On two relator groups. In Paul Latiolais, editor, Topology and Combinatorial Group Theory, Proceedings of the Fall Foliage Topology Seminars held in New Hampshire 1985-1988, volume 1440 of Lecture Notes Math., pages 1–21. SpringerVerlag Berlin Heidelberg, 1990. [6] Y.G. Baik, William A. Bogley, and Stephen J. Pride. On the asphericity of length four relative group presentations. Int. J. Algebra Comput., 7(3):277–312, 1997. [7] V. G. Bardakov and A.Yu. Vesnin. A generalization of Fibonacci groups. Algebra Logika, 42(2):131–160, 2003. Translation in Algebra Logic 42(2):73–91 (2003). [8] A. U. Besche, B. Eick, and E.A. O’Brien. The SmallGroups Library – a GAP package, 2002. [9] William A. Bogley. An identity theorem for multi-relator groups. Math. Proc. Camb. Philos. Soc., 109(2):313–321, 1991. [10] William A. Bogley. On shift dynamics for cyclically presented groups. J. Algebra, 418:154–173, 2014. [11] William A. Bogley and Forrest W. Parker. Cyclically presented groups with length four positive relators. J. Group Theory, 21:911–948, 2018. [12] William A. Bogley and Stephen J. Pride. Aspherical relative presentations. Proc. Edin. Math. Soc., 35(1):1–39, 1992. [13] William A. Bogley and Steve J. Pride. Calculating generators of π2 . In C. HogAngeloni, W. Metzler, and A.J. Sieradski, editors, Two-dimensional homotopy and combinatorial group theory, volume 197 of London Math. Soc. Lecture Note Ser., pages 157–188. Cambridge: Cambridge University Press, 1993.

Bogley, Edjvet, Williams: Aspherical Relative Presentations Again

197

[14] William A. Bogley and Gerald Williams. Efficient finite groups arising in the study of relative asphericity. Math. Z., 284(1-2):507–535, 2016. [15] William A. Bogley and Gerald Williams. Coherence, subgroup separability, and metacyclic structures for a class of cyclically presented groups. J. Algebra, 480:266– 297, 2017. [16] Kenneth S. Brown. Cohomology of groups, volume 87 of Graduate Text Math. Springer New York, 1982. [17] Kenneth S. Brown. Lectures on the cohomology of groups. In Cohomology of groups and algebraic K-theory. Selected papers of the international summer school on cohomology of groups and algebraic K-theory, Hangzhou, China, July 1–3, 2007, pages 131–166. Somerville, MA: International Press; Beijing: Higher Education Press, 2010. [18] A. Cavicchioli, F. Hegenbarth, and A. C. Kim. A geometric study of Sieradski groups. Algebra Colloq., 5(2):203–217, 1998. [19] Alberto Cavicchioli, E. A. O’Brien, and Fulvia Spaggiari. On some questions about a family of cyclically presented groups. J. Algebra, 320(11):4063–4072, 2008. [20] Christopher P. Chalk. Fibonacci groups with aspherical presentations. Commun. Algebra, 26(5):1511–1546, 1998. [21] Ian M. Chiswell, Donald J. Collins, and Johannes Huebschmann. Aspherical group presentations. Math. Z., 178:1–36, 1981. [22] J. M. Corson and B. Trace. Diagrammatically reducible complexes and Haken manifolds. J. Aust. Math. Soc., Ser. A, 69(1):116–126, 2000. [23] Peter J. Davidson. On the asphericity of a family of relative group presentations. Int. J. Algebra Comput., 19(2):159–189, 2009. [24] Eldon Dyer and A. T. Vasquez. Some small aspherical spaces. J. Aust. Math. Soc., 16:332–352, 1973. [25] Martin Edjvet. Equations over groups and a theorem of Higman, Neumann and Neumann. Proc. Lond. Math. Soc. (3), 63(3):563–589, 1991. [26] Martin Edjvet. Solutions of certain sets of equations over groups. In Groups St Andrews 1989, Volume 1, number 159 in Lecture Note Ser., pages 105–123. London Math. Soc., Cambridge University Press, 1991. [27] Martin Edjvet. On the asphericity of one-relator relative presentations. Proc. R. Soc. Edinb., Sect. A, 124(4):713–728, 1994. [28] Martin Edjvet and Arye Juhasz. The infinite Fibonacci groups and relative asphericity. Trans. Lond. Math. Soc., 4(1):148–218, 2017. [29] Max Forester and Colin Rourke. A fixed-point theorem and relative asphericity. Enseign. Math. (2), 51(3-4):231–237, 2005. [30] Michael H. Freedman. Remarks on the solution of first degree equations in groups. In K. C. Millet, editor, Algebraic and Geometric Topology Proceedings of a Symposium held at Santa Barbara in honor of Raymond L. Wilder, July 25–29, 1977, volume 664 of Lect. Notes Math., pages 87–93. Springer, 1978. [31] The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.8.7, 2017. [32] S. M. Gersten. On certain equations over torsion free groups. [33] S. M. Gersten. Reducible diagrams and equations over groups. In S. M. Gersten, editor, Essays in Group Theory, volume 8 of Publ. Math. Sci. Res. Inst., pages 15–73. Springer, New York, 1987. [34] M. Gerstenhaber and O. S. Rothaus. The solution of sets of equations in groups. Proc. Natl. Acad. Sci. USA, 48:1531–1533, 1962. [35] N. D. Gilbert and James Howie. LOG groups and cyclically presented groups. J. Algebra, 174(1):118–131, 1995.

198

Bogley, Edjvet, Williams: Aspherical Relative Presentations Again

[36] Allen Hatcher. Algebraic topology. Cambridge: Cambridge University Press, 2002. [37] H. Helling, A. C. Kim, and J. L. Mennicke. A geometric study of Fibonacci groups. J. Lie Theory, 8(1):1–23, 1998. [38] James Howie. On pairs of 2-complexes and systems of equations over groups. J. Reine Angew. Math., 324:165–174, 1981. [39] James Howie. The solution of length three equations over groups. Proc. Edinb. Math. Soc., II. Ser., 26:89–96, 1983. [40] James Howie. How to generalize one-relator group theory. In S. M. Gersten and John R. Stallings, editors, Combinatorial group theory and topology, volume 111 of Annals of Mathematics Studies, pages 53–78. Princeton University Press, 1987. [41] James Howie. The quotient of a free product of groups by a single high-powered relator. I: Pictures. Fifth and higher powers. Proc. Lond. Math. Soc. (3), 59(3):507– 540, 1989. [42] James Howie. Nonsingular systems of two length three equations over a group. Math. Proc. Camb. Philos. Soc., 110(1):11–24, 1991. [43] James Howie and V. Metaftsis. On the asphericity of length five relative group presentations. Proc. Lond. Math. Soc. (3), 82(1):173–194, 2001. [44] James Howie and Hans Rudolf Schneebeli. Permutation modules and projective resolutions. Comment. Math. Helv., 56:447–464, 1981. [45] James Howie and Gerald Williams. Fibonacci type presentations and 3-manifolds. Topology Appl., 215:24–34, 2017. [46] Johannes Huebschmann. Cohomology theory of aspherical groups and of small cancellation groups. J. Pure Appl. Algebra, 14:137–143, 1979. [47] S. V. Ivanov. An asphericity conjecture and Kaplansky problem on zero divisors. J. Algebra, 216(1):13–19, 1999. [48] D. L. Johnson. Topics in the Theory of Group Presentations, volume 42 of London Math. Soc. Lecture Note Ser. Cambridge: Cambridge University Press, 1980. [49] Irving Kaplansky. Fields and rings. Chicago Lectures in Mathematics. Chicago, IL: University of Chicago Press, 2nd edition, 1972. [50] Seong Kun Kim. On the asphericity of certain relative presentations over torsion-free groups. Int. J. Algebra Comput., 18(6):979–987, 2008. [51] Seong Kun Kim. On the asphericity of length-6 relative presentations with torsion-free coefficients. Proc. Edinb. Math. Soc., II. Ser., 51(1):201–214, 2008. [52] Anton A. Klyachko. A funny property of sphere and equations over groups. Comm. Algebra, 21(7):2555–2575, 1993. [53] Anton A. Klyachko. The Kervaire-Laudenbach conjecture and presentations of simple groups. Algebra Logika, 44(4):399–437, 2005. [54] Ian J. Leary. Asphericity and zero divisors in group algebras. J. Algebra, 227(1):362– 364, 2000. [55] Roger C. Lyndon. Cohomology theory of groups with a single defining relation. Ann. Math. (2), 52:650–665, 1950. [56] Roger C. Lyndon and Paul E. Schupp. Combinatorial group theory. Berlin: Springer, reprint of the 1977 edition, 2001. [57] Wilhelm Magnus, Abraham Karrass, and Donald Solitar. Combinatorial group theory. Presentations of groups in terms of generators and relations. Dover Books on Advanced Mathematics. Dover Publications, 2nd edition, 1976. [58] Kirk M. McDermott. Topological and Dynamical Properties of Cyclically Presented Groups. PhD thesis, Oregon State University, 2017. [59] V. Metaftsis. On the asphericity of relative group presentations of arbitrary length. Int. J. Algebra Comput., 13(3):323–339, 2003.

Bogley, Edjvet, Williams: Aspherical Relative Presentations Again

199

[60] M. S. Montgomery. Left and right inverses in group algebras. Bull. Am. Math. Soc., 75:539–540, 1969. [61] B. H. Neumann. Adjunction of elements to groups. J. Lond. Math. Soc., 18:4–11, 1943. [62] R. W. K. Odoni. Some Diophantine problems arising from the theory of cyclicallypresented groups. Glasg. Math. J., 41(2):157–165, 1999. [63] A. Yu. Ol’shanskij. On a geometric method in the combinatorial group theory. In Proc. Int. Congr. Math., Warszawa 1983, volume 1, pages 415–424. Warszawa: PWNPolish Scientific Publishers, 1984. [64] A. Yu. Ol’shanskij. Geometry of defining relations in groups. Dordrecht etc.: Kluwer Academic Publishers, 1991. [65] M. I. Prishchepov. Asphericity, atoricity and symmetrically presented groups. Comm. Algebra, 23(13):5095–5117, 1995. [66] John G. Ratcliffe. Euler characteristics of 3-manifold groups and discrete subgroups of SL(2, C). J. Pure Appl. Algebra, 44:303–314, 1987. [67] Allan J. Sieradski. A coloring test for asphericity. Q. J. Math., Oxf. II. Ser., 34:97– 106, 1983. [68] Allan J. Sieradski. Combinatorial squashings, 3-manifolds, and the third homology of groups. Invent. Math., 84:121–139, 1986. [69] Allan J. Sieradski. Algebraic topology for two dimensional complexes. In C. HogAngeloni, W. Metzler, and A.J. Sieradski, editors, Two-dimensional Homotopy and Combinatorial Group Theory, number 197 in London Math. Soc. Lecture Note Ser., pages 51–96. Cambridge: Cambridge University Press, 1993. [70] C. T. C. Wall. Rational Euler characteristics. Proc. Camb. Philos. Soc., 57:182–184, 1961. [71] J. H. C. Whitehead. Combinatorial homotopy. II. Bull. Am. Math. Soc., 55:453–496, 1949. [72] Gerald Williams. The aspherical Cavicchioli-Hegenbarth-Repovˇs generalized Fibonacci groups. J. Group Theory, 12(1):139–149, 2009. [73] Gerald Williams. Unimodular integer circulants associated with trinomials. Int. J. Number Theory, 6(4):869–876, 2010. [74] Gerald Williams. Groups of Fibonacci type revisited. Int. J. Algebra Comput., 22(8), 2012.

SIMPLE GROUPS, GENERATION AND PROBABILISTIC METHODS TIMOTHY C. BURNESS∗ ∗

School of Mathematics, University of Bristol, Bristol BS8 1TW, UK Email: [email protected]

Abstract It is well known that every finite simple group can be generated by two elements and this leads to a wide range of problems that have been the focus of intensive research in recent years. In this survey article we discuss some of the extraordinary generation properties of simple groups, focussing on topics such as random generation, (a, b)-generation and spread, as well as highlighting the application of probabilistic methods in the proofs of many of the main results. We also present some recent work on the minimal generation of maximal and second maximal subgroups of simple groups, which has applications to the study of subgroup growth and the generation of primitive permutation groups.

1

Introduction

In this survey article we will discuss some of the remarkable generation properties of finite simple groups. Our starting point is the fact that every finite simple group can be generated by just two of its elements (this is essentially a theorem of Steinberg [70], and the proof requires the Classification of Finite Simple Groups). This leads naturally to a wide range of interesting questions concerning the abundance of generating pairs and their distribution across the group, which have been intensively studied in recent years. Our goal in Sections 2 and 3 is to survey some of the main results and open problems. Another one of our aims is to highlight the central role played by probabilistic methods. In some instances, the given result may already be stated in probabilistic terms (for example, it may refer to the probability that two randomly chosen elements in a group form a generating pair). However, we will see that probabilistic techniques have also been used in an essential way to prove entirely deterministic statements. A striking example is given by Guralnick and Kantor’s proof of the so-called 32 -generation property for simple groups, which we will discuss in Section 3, together with some far-reaching generalisations. Our understanding of the subgroups of finite simple groups has advanced greatly in recent years. In particular, many results on the generation of simple groups rely on powerful subgroup structure theorems such as the O’Nan–Scott Theorem for alternating groups and Aschbacher’s theorem for classical groups. In a different direction, it is natural to consider the generation properties of the subgroups themselves, such as maximal and second maximal subgroups that are located at the top of the subgroup lattice. Indeed, we can seek to understand the extent to which

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some of the remarkable results for simple groups extend to these subgroups (with suitable modifications, if necessary). The study of problems of this nature was recently initiated through joint work with Liebeck and Shalev and we will discuss some of the main results in Section 4. The coverage of this article is based on the content of my one-hour lecture at the Groups St Andrews conference, which was hosted by the University of Birmingham in August 2017. It is a pleasure to thank the organisers for inviting me to give this lecture and for planning and delivering a very interesting and inspiring meeting. There is a vast literature on the generation of simple groups and so I have had to be very selective in choosing the main topics for this article, which is rather biased towards my own tastes and interests. There are many other excellent survey articles on related topics, which an interested reader may wish to consult. For example, Shalev has written several interesting articles on the use of probabilistic methods in finite group theory, which discuss applications to generation problems and much more (see [68], for example). Liebeck’s survey article [40] on probabilistic group theory provides an excellent account of some of the more recent developments. Let us say a few words on the notation used in this article. In general, our notation is all fairly standard (and new notation will be defined when needed). It might be helpful to point out that we use the notation of [37] for simple groups. For example, we will write Ln (q) = PSLn (q) and Un (q) = PSUn (q) for linear and unitary groups, and we use PΩ+ n (q), etc., for simple orthogonal groups (this differs from the notation used in the Atlas [19]). We will also write Zn for a cyclic group of order n. Finally, I would like to thank Scott Harper and an anonymous reviewer for helpful comments on an earlier version of this article.

2

Generation properties of simple groups

Let G be a finite group and let d(G) = min{|S| : G = S } be the minimal number of generators for G. We will say that G is n-generated if d(G) is at most n. In this section we will focus on the generation properties of simple groups, which is an area of research with a long and rich history. Here the most well-known result is the fact that every finite simple group can be generated by two elements. Theorem 2.1 Every finite simple group is 2-generated. The proof relies on the Classification of Finite Simple Groups. First observe that the alternating groups are straightforward. For example, it is an easy exercise to show that

(1, 2, 3), (1, 2, . . . , n) n odd An = (1, 2, 3), (2, 3, . . . , n) n even.

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In [70], Steinberg presents explicit generating pairs for each simple group of Lie type. For instance, L2 (q) = xZ, yZ , where Z = Z(SL2 (q)) and     α 0 −1 1 x= , y= −1 0 α −1 0 with F× q = α . In [3], Aschbacher and Guralnick complete the proof of the theorem by showing that every sporadic group is 2-generated. In view of Theorem 2.1, there are many natural extensions and variations to consider. For example: 1. How abundant are generating pairs in a finite simple group? 2. Can we find generating pairs of prescribed orders? 3. Does every non-identity element belong to a generating pair? As we shall see, problems of this flavour have been the focus of intensive research in recent years, with probabilistic methods playing a central role in the proofs of many of the main results. 2.1

Random generation

Let G be a finite group, let k be a positive integer and let Pk (G) =

|{(x1 , . . . , xk ) ∈ Gk : G = x1 , . . . , xk }| |G|k

be the probability that k randomly chosen elements generate G. For a simple group G, Theorem 2.1 implies that P2 (G) > 0 and it is natural to consider the asymptotic behaviour of P2 (G) with respect to |G|. This is an old problem, which can be traced all the way back to a conjecture of Netto in 1882. In [61], Netto writes “If we arbitrarily select two or more substitutions of n elements, it is to be regarded as extremely probable that the group of lowest order which contains these is the symmetric group, or at least the alternating group.” In other words, Netto is predicting that P2 (An ) → 1 as n tends to infinity. This conjecture was proved by Dixon [21] in a highly influential paper published in 1969, which relies in part on the pioneering work of Erd˝ os and Tur´ an [24] on statistical properties of symmetric groups. In the same paper, Dixon makes the bold conjecture that all finite simple groups are strongly 2-generated in the sense of Netto. Conjecture 2.2 (Dixon, 1969) Let (Gn ) be any sequence of finite simple groups such that |Gn | tends to infinity with n. Then P2 (Gn ) → 1 as n → ∞. The proof of Dixon’s conjecture was eventually completed in the 1990s. In [35], Kantor and Lubotzky establish the result for classical groups and low rank exceptional groups, and the remaining groups of Lie type were handled by Liebeck and Shalev [43]. The proof is based on the following elementary observations.

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Let G be a finite group and let M be the set of maximal subgroups of G. Set  ζG (s) = |G : H|−s (1) H∈M

for a real number s > 0. For randomly chosen elements x, y ∈ G, observe that G = x, y if and only if x, y ∈ H for some H ∈ M. Since |G : H|−2 is the probability that these random elements are both contained in H, it follows that  1 − P2 (G)  |G : H|−2 = ζG (2). H∈M

Now assume that G is a finite simple group of Lie type. By carefully studying M, using powerful results on the subgroup structure of these groups, such as Aschbacher’s theorem [1] for classical groups, one can show that ζG (2) → 0 as |G| tends to infinity. Therefore P2 (G) → 1 and Dixon’s conjecture follows. It is interesting to note that this probabilistic argument shows that every sufficiently large finite simple group of Lie type is 2-generated, without the need to explicitly construct a pair of generators. Numerous extensions have since been established. For example, the following striking result is [58, Theorem 1.1]. Theorem 2.3 We have P2 (G)  53/90 for every non-abelian finite simple group G, with equality if and only if G = A6 . It turns out that convergence in Conjecture 2.2 is rather rapid and strong bounds on P2 (G) have been established. The most general result is a theorem of Liebeck and Shalev [44, Theorem 1.6], which states that there are absolute constants c1 , c2 > 0 such that c2 c1  P2 (G)  1 − 1− m(G) m(G) for all finite simple groups G, where m(G) denotes the minimal index of a proper subgroup of G. We refer the reader to [28, Table 4] for a convenient list of the precise values of m(G) when G is a simple group of Lie type. For an alternating group An with n  5 we have m(An ) = n and explicit bounds on P2 (An ) are available. For instance, [60, Theorem 1.1] gives 1−

1 8.8 1 0.93 − 2  P2 (An )  1 − − 2 n n n n

for all n  5. 2.2

(a, b)-generation

Another interesting refinement of Theorem 2.1 is to ask if it is possible to find a pair of generators of prescribed orders. With this in mind, for positive integers a and b, let us say that a finite group G is (a, b)-generated if G = x, y with |x| = a and |y| = b. It is natural to assume that both a and b are primes, at least one of which is odd (since any group generated by two involutions is dihedral). Here

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the special case (a, b) = (2, 3) is particularly interesting because a group is (2, 3)generated if and only if it is a quotient of the modular group PSL2 (Z) ∼ = Z2  Z3 . The (2, 3)-generation problem for simple groups has been widely studied for more than a century and one of the main results is the following. Theorem 2.4 All sufficiently large non-abelian finite simple groups are (2, 3)generated, with the exception of PSp4 (2f ), PSp4 (3f ) and 2 B2 (q). This follows from an old theorem of Miller [59] for alternating groups, which reveals that An is (2, 3)-generated unless n ∈ {6, 7, 8}. The result for classical groups was proved by Liebeck and Shalev [45] and the exceptional groups were handled by L¨ ubeck and Malle in [49]. More precisely, the latter paper shows that every simple exceptional group of Lie type is (2, 3)-generated, except for G2 (2) ∼ = U3 (3) and of course the Suzuki groups 2 B2 (q), which do not contain elements of order 3 (Suzuki [71] showed that these groups are (2, 5)-generated). For completeness, let us record that every sporadic simple group is (2, 3)-generated, except for M11 , M22 , M23 and McL (see [74]). For classical groups, Liebeck and Shalev adopt a probabilistic approach in [45], which uses several results on the maximal subgroups of classical groups, such as Aschbacher’s theorem [1] and its extensions. As one might expect, detailed information on the conjugacy classes of elements of order 2 and 3 also plays an important role in the proofs. In [49], L¨ ubeck and Malle adopt rather different techniques to study the (2, 3)-generation of exceptional groups. Indeed, their main methods are character-theoretic, using the Deligne–Lusztig theory for reductive groups over finite fields. Let us briefly sketch the main ideas in [45]. For positive integers a and b, let Pa,b (G) be the probability that G is generated by randomly chosen elements of order a and b, so G is (a, b)-generated if and only if Pa,b (G) > 0. It is easy to see that  ia (H)ib (H) 1 − Pa,b (G)  , (2) ia (G)ib (G) H∈M

where M is the set of maximal subgroups of G as before, and im (X) is the number of elements of order m in X. By carefully estimating i2 (H) and i3 (H) for H ∈ M, Liebeck and Shalev show that there is an absolute constant c such that   i2 (H)i3 (H) < c|G : H|−66/65 = c · ζG (66/65) i2 (G)i3 (G)

H∈M

H∈M

for any finite simple classical group G = PSp4 (q), where ζG (s) is the zeta function in (1) (see [45, Theorems 2.2 and 2.3]). The result now follows from [45, Theorem 2.1], which states that for any s > 1, ζG (s) → 0 as |G| → ∞ (note that ζG (1) is equal to the number of conjugacy classes of maximal subgroups of G, which tends to infinity with |G|). Moreover, by combining this result with [45, Proposition 6.3], we get the following natural analogue of Conjecture 2.2 for the (2, 3)-generation of classical groups.

Burness: Simple groups, generation and probabilistic methods Theorem 2.5 For finite simple ⎧ ⎨ 0 1 P2,3 (G) → ⎩ 2 1

205

classical groups G, as |G| → ∞ we have if G = PSp4 (pf ) with p = 2 or 3 if G = PSp4 (pf ) with p = 2, 3 otherwise.

Using a similar approach, the main theorem of [46] shows that if a and b are primes, not both equal to 2, then Pa,b (G) → 1 as |G| → ∞, for all simple classical groups G of sufficiently large rank (a sufficient bound on the rank can be given as a function of a and b). Remark 2.6 A complete classification of the (2, 3)-generated finite simple groups remains out of reach, but there has been significant progress by Di Martino, Pellegrini, Tamburini, Vavilov and others, using constructive methods. For example, Pellegrini [64] has very recently resolved the (2, 3)-generation problem for the linear groups Ln (q); the only exceptions arise when (n, q) ∈ {(2, 9), (3, 4), (4, 2)}, all of which are (2, 5)-generated. In their recent survey article [65], Pellegrini and + Tamburini make the interesting observation that Ω+ 8 (2) and PΩ8 (3) are the only known simple classical groups with natural module of dimension n  8 that are not (2, 3)-generated. To conclude this section, let us briefly discuss the more general (2, r)-generation problem. By a theorem of Malle, Saxl and Weigel [53, Theorem B], every finite simple group G is (2, r)-generated for some integer r  3. In fact, for G = U3 (3), it is proved that G is generated by an involution and a strongly real element (that is, an element x so that x−1 = y −1 xy for some involution y), which immediately implies that G is generated by 3 involutions (one can show that 4 involutions are needed for U3 (3)). The following refinement of King [36] shows that r can be taken to be a prime. Theorem 2.7 Let G be a non-abelian finite simple group. Then there exists a prime r such that G is (2, r)-generated. Once again, the proof uses probabilistic methods and we give a brief sketch of the main steps. In view of earlier work, we may assume that G is a classical group over Fq , with natural module of dimension n. By applying the bound in (2) (with a = 2 and b = 5), King shows that the symplectic groups PSp4 (2f ) and PSp4 (3f ) are (2, 5)-generated. By combining this observation with previous results in the literature, the problem can be reduced to classical groups with n  8. To tackle these groups, we need to recall the notion of a primitive prime divisor. Definition 2.8 For integers q, e  2, a prime divisor r of q e − 1 is a primitive prime divisor (ppd for short) if r does not divide q i − 1 for each 1  i < e. By a classical theorem of Zsigmondy [75], such a prime r exists unless (q, e) = (2a − 1, 2) or (2, 6). Let r be a ppd of q e − 1, where e = e(n) is maximal with respect to the condition that r divides |G|. For instance, e = n if G = Ln (q) or

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PSpn (q), and e = n − 2 if G = PΩ+ n (q). Let x ∈ G be an element of order r and let M(x) be the set of maximal subgroups of G containing x. For (2, r)-generation, it suffices to show that P2 (G, x) > 0, where P2 (G, x) is the probability that x and a randomly chosen involution generate G. Now 1 − P2 (G, x) 

 H∈M(x)

i2 (H) i2 (G)

(3)

and this essentially reduces the argument to determining M(x) and then counting the involutions in each H ∈ M(x). By choosing r to be a ppd of q e − 1 with e > n/2, we can appeal to [30], which uses Aschbacher’s theorem [1] to determine the maximal subgroups of G containing elements of order r. It follows that the subgroups in M(x) are very restricted and this makes it easier to estimate the upper bound in (3). This approach is effective in almost all cases, with only a handful of low-dimensional groups requiring further attention (see [36, Section 7]). Notice that King’s proof does not yield an absolute bound on the prime r in the statement of the theorem (indeed, r tends to infinity with the rank of the group). However, it is natural to ask if there is an absolute constant R such that every finite simple group is (2, r)-generated for some prime r  R. In view of the above results, it is not difficult to show that r  5 if G is an alternating or sporadic group, and King’s proof shows that r  7 if G is a classical group with natural module of dimension n  7 (the group U3 (3) is neither (2, 3) nor (2, 5)-generated). The bound r  7 also holds for exceptional groups of Lie type. This leads us naturally to the following conjecture of Conder, which is still open. Conjecture 2.9 (Conder, 2015) Every non-abelian finite simple group is (2, r)generated for some r ∈ {3, 5}, except for U3 (3), which is (2, 7)-generated. 2.3

Triangle generation

Let a, b and c be positive integers with a  b  c. We say that a group G is (a, b, c)generated if G = x, y for elements x, y ∈ G such that |x|, |y| and |xy| divide a, b and c, respectively. This is equivalent to the condition that G is a quotient of the triangle group Ta,b,c = x, y, z | xa = y b = z c = xyz = 1 . The problem of determining the finite simple quotients of triangle groups has attracted significant attention for more than a century. One of the main motivations stems from a famous theorem of Hurwitz from 1893, which states that |Aut(S)|  84(g − 1) for any compact Riemann surface S of genus g  2, with equality if and only if Aut(S) is a (2, 3, 7)-group (these groups are called Hurwitz groups). There has been substantial progress towards a classification of simple Hurwitz groups, but this remains an open problem (see [18] for a nice survey of results). One of the

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highlights is the following theorem of Conder [17], which settles a conjecture of Higman from the 1960s asserting that all but finitely many alternating groups are Hurwitz. Theorem 2.10 The alternating group An is a Hurwitz group for all n  168, and for all but 64 integers in the range 3  n  167. For the remainder of this section, we will discuss some more recent results concerning the triangle generation of finite simple groups and related problems. Let G be a simple algebraic group over an algebraically closed field K of characteristic p > 0. For a fixed triple (a, b, c) of integers, it is natural to ask if there are any values of r such that the corresponding finite quasisimple group G(pr ) is (a, b, c)-generated. Here we may assume that (a, b, c) is hyperbolic, which means that 1 1 1 + + < 1. a b c Indeed, if this condition is not satisfied, then either Ta,b,c is soluble, or (a, b, c) = (2, 3, 5) and Ta,b,c ∼ = A5 . In [52], Macbeath proves that L2 (pr ) is Hurwitz if and only if r = 1 and p ≡ 0, ±1 (mod 7), or r = 3 and p ≡ ±2, ±3 (mod 7). This is extended by Marion in [55, Corollary 1], which states that if (a, b, c) is a hyperbolic triple of primes and p is any prime number, then L2 (pr ) is (a, b, c)-generated if and only if pr is the smallest power of p such that lcm(a, b, c) divides |L2 (pr )| (in particular, r is unique). To shed further light on Marion’s result for L2 (pr ), we need some additional terminology. For a positive integer m, let jm (G) be the dimension of the subvariety of G of elements of order dividing m. Let us say that a triple (a, b, c) of positive integers is rigid for G if ja (G) + jb (G) + jc (G) = 2 dim G.

(4)

We can now state the following conjecture (see [56, p.621]). Conjecture 2.11 (Marion, 2010) Fix a prime p and let G be a simple algebraic group over an algebraically closed field of characteristic p > 0. If (a, b, c) is a rigid hyperbolic triple of primes for G, then there are only finitely many positive integers r such that G(pr ) is (a, b, c)-generated. In the special case G = PSL2 (K) we have dim G = 3 and jm (G) = 2 for all m  2, so every triple is rigid for G and thus the conclusion of the conjecture is in agreement with [55, Corollary 1], as stated above. Significant progress towards a proof of the conjecture is made in [56], where Marion reduces the problem to a handful of cases with G = Sp2m (K) (for m  13), PSp4 (K) or G2 (K). To do this, one first determines the rigid hyperbolic triples of primes for G, which is a relatively straightforward exercise using the known dimensions of conjugacy

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classes of elements of prime order in simple algebraic groups. The rigidity condition in (4) is highly restrictive. For instance, if G is an exceptional group, then G = G2 (K) with (a, b, c) = (2, 5, 5) is the only possibility. To complete the reduction, the main step is to eliminate a handful of linear groups G = SLn (K). Here the main tool is a well-known theorem of Scott [67], which is used to show that GLn (K) has only finitely many orbits on the set {(x, y, z) ∈ G3 : xa = y b = z c = 1, xyz = 1, x, y irreducible}, where a subgroup of G is said to be irreducible if it acts irreducibly on the natural module for G. Up to conjugacy in GLn (K), it follows that there are only finitely many irreducible (a, b, c)-generated subgroups of G, which immediately gives the desired result in these cases. Using a completely different approach, Larsen, Lubotzky and Marion [38] apply tools from deformation theory to study a generalised version of Conjecture 2.11, where the prime condition on the triple (a, b, c) is dropped. The main result is [38, Theorem 1.7], which establishes this extended form of the conjecture unless p divides abcd, where d is the determinant of the Cartan matrix of G. This result has recently been pushed further in [34], which proves the extended conjecture for all simple groups G(pr ). As in [56], the approach in [34] relies heavily on the classification of hyperbolic rigid triples (with no prime conditions) and most of the work involves the case G = PSp4 (K) with (a, b) = (3, 3), which requires special attention and different techniques. Conjecture 2.11 for quasisimple groups is still open, even for prime triples. It is also worth noting that the converse to the conjecture is false. For example, (2, 3, 7) is non-rigid for G = SL7 (K), but SL7 (q) is never a Hurwitz group for any prime power q (see [56, p.623] for further details and examples). Natural extensions of triangle generation can be studied by observing that every hyperbolic triangle group Ta,b,c is a special type of Fuchsian group. This broader family of groups arises naturally in geometry and combinatorial group theory (formally, a Fuchsian group is a finitely generated non-elementary discrete group of isometries of the hyperbolic plane H2 ). An orientation-preserving Fuchsian group Γ has a rather simple presentation, with generators a1 , b1 , . . . , ag , bg , x1 , . . . , xd , y1 , . . . , ys , z1 , . . . , zt and relations md 1 xm 1 = · · · = xd = 1, x1 · · · xd y1 · · · ys z1 · · · zt [a1 , b1 ] · · · [ag , bg ] = 1

with g, d, s, t  0 and mi  2 for all i. Here g  0 is called the genus of Γ (the non-orientation-preserving groups admit a similar presentation). In this setting, a triangle group corresponds to the situation where g = s = t = 0 and d = 3, so it is natural to extend the notion of triangle generation by considering the finite quotients of arbitrary Fuchsian groups. A number of remarkable results in this direction have been established in recent years. For instance, Everitt [25] has shown that if Γ is an oriented Fuchsian group,

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then all but finitely many alternating groups are quotients of Γ. This establishes a conjecture of Higman (in the oriented case), which generalises Conder’s theorem on Hurwitz groups (see Theorem 2.10). Everitt’s approach in [25] builds on the coset-diagram methodology developed by Higman and Conder. By applying very different methods, using a combination of character-theoretic and probabilistic tools, Liebeck and Shalev prove the following theorem, which settles Higman’s conjecture in full generality (see [47, Theorem 1.7]). In the statement, we use the notation Homtrans (Γ, An ) = {ϕ ∈ Hom(Γ, An ) : ϕ(Γ) is transitive}. Theorem 2.12 Let Γ be a Fuchsian group. Then the probability that a random homomorphism in Homtrans (Γ, An ) is an epimorphism tends to 1 as n → ∞. Numerous extensions are pursued in [48], where An is replaced by a different simple group. For example, the following striking result is [48, Theorem 1.6]. Theorem 2.13 Let Γ be a Fuchsian group of genus g  2 (g  3 if non-oriented), and let G be a finite simple group. Then the probability that a randomly chosen homomorphism in Hom(Γ, G) is an epimorphism tends to 1 as |G| → ∞. The condition on the genus here is essential, since there are Fuchsian groups of genus 0 or 1 which do not have all large enough finite simple groups as quotients. Indeed, we have already seen that if (a, b, c) is a hyperbolic triple of primes then L2 (pr ) is a quotient of Ta,b,c for just one value of r. For arbitrary Fuchsian groups, the following conjecture is still open (see [48, p.323]). Conjecture 2.14 (Liebeck & Shalev, 2005) For any Fuchsian group Γ there is an integer f (Γ), such that if G is a finite simple classical group of rank at least f (Γ), then the probability that a randomly chosen homomorphism in Hom(Γ, G) is an epimorphism tends to 1 as |G| → ∞.

3

Spread

In the previous section, we highlighted several strong 2-generation properties of simple groups, which can be viewed as far-reaching generalisations of Theorem 2.1. In this section we study the notions of spread and uniform spread, which provide yet another way to demonstrate the effortless 2-generation of simple groups. 3.1

Definitions

We begin with the following definition, which was introduced by Brenner and Wiegold [8] in the 1970s. Definition 3.1 Let G be a finite group and let k be a positive integer. Then G has spread k if for any non-identity elements x1 , . . . , xk ∈ G there exists y ∈ G such that G = xi , y for all i. We say that G is 32 -generated if it has spread 1.

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One of the main motivations stems from earlier work of Binder [4], who proved that Sn has spread 2 for all n  5 (in fact, there is an even earlier result of Piccard [66] from the 1930s, which states that both An and Sn are 32 -generated if n  5). Notice that every cyclic group has spread k for all k ∈ N, so for the remainder of Section 3 we will assume that G is non-cyclic. Set s(G) = max{k ∈ N0 : G has spread k}. In practice, it can often be more convenient to work with the more restrictive notion of uniform spread, which was formally introduced much more recently in [10]. Definition 3.2 A finite group G has uniform spread k if there exists a fixed conjugacy class C of G such that for any non-identity elements x1 , . . . , xk ∈ G there exists y ∈ C such that G = xi , y for all i. For a non-cyclic group G, set u(G) = max{k ∈ N0 : G has uniform spread k} and observe that u(G)  s(G) < |G| − 1. Note that the first inequality can be strict; for example, u(S6 ) = 0 and s(S6 ) = 2. 3.2

The spread of simple groups

In [8], Brenner and Wiegold extend the earlier work of Binder and Piccard by investigating the spread of various families of simple groups. Among several interesting results, they prove that s(A2n ) = 4 for all n  4 and they show that ⎧ ⎨ q − 1 if q ≡ 1 (mod 4) q − 4 if q ≡ 3 (mod 4) s(L2 (q)) = ⎩ q − 2 if q is even for q  11 (see [8, Theorems 3.10 and 4.02]). In particular, the spread of a finite simple group can be arbitrarily large. They also observe that the spread of odd degree alternating groups is radically different. For instance, [8, Theorem 4.01] states that 6098892799  s(A19 )  6098892803.

(5)

Later work by Guralnick and Shalev [31] shows that s(Ap ) tends to infinity with p when p is a prime number. More generally, [31, Theorem 1.1] implies that if (Gi ) is a sequence of alternating groups such that Gi = Ani and ni tends to infinity with i, then s(Gi ) → ∞ if and only if f (ni ) → ∞, where f (ni ) is the smallest prime divisor of ni . In Steinberg’s original paper [70], where he presents a generating pair for each simple group of Lie type, he suggests that these groups may have the much stronger 3 2 -generation property (Steinberg was familiar with Piccard’s result for alternating groups). Steinberg’s prediction was eventually verified almost 40 years later (in a stronger form) by Stein [69], and independently by Guralnick and Kantor [29].

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Theorem 3.3 If G is a non-abelian finite simple group, then u(G)  1. Stronger results are established in [29], which turn out to be important for subsequent improvements of the bound in Theorem 3.3. More precisely, it is shown that there is a conjugacy class C of G such that each non-identity element of G generates G with at least 1/10 of the elements in C, and they also establish some related results for almost simple groups. In later work [10], Breuer, Guralnick and Kantor show that the constant 1/10 can be improved to 13/42 (for G = Ω+ 8 (2), this is best possible), and by excluding a short list of known groups, 1/10 can be replaced by 2/3. As we shall see below, the fact that the above fraction 1/10 can be replaced by 2/3 in almost all cases is the key ingredient in the proof of the following theorem, which is the main result on the spread of simple groups (see [10, Corollary 1.3]). Theorem 3.4 Let G be a non-abelian finite simple group. Then u(G)  2, with equality if and only if G ∈ {A5 , A6 , Ω+ 8 (2), Sp2m (2) (m  3)}. The proof of Theorem 3.4 uses probabilistic methods, based on fixed point ratio estimates. To describe the main ideas, we need some notation. Let G be a finite group. For x, y ∈ G, let P(x, y) =

|{z ∈ y G : G = x, z }| |y G |

be the probability that x and a randomly chosen conjugate of y generate G. Set Q(x, y) = 1 − P(x, y). For a subgroup H  G and element x ∈ G, let fpr(x, G/H) =

|xG ∩ H| |xG |

be the fixed point ratio of x. This is the proportion of fixed points of x under the natural action of G on the set of cosets of H in G. Notice that fpr(x, G/H)  fpr(xm , G/H) for all m ∈ N. Lemma 3.5 Suppose there exists an element y ∈ G and a positive integer k such that Q(x, y) < 1/k for all 1 = x ∈ G. Then u(G)  k. Proof Let x1 , . . . , xk ∈ G be non-identity elements and set E = E1 ∩ · · · ∩ Ek , where Ei is the event that G = xi , z for a randomly chosen conjugate z ∈ y G . Then ¯ = 1 − P(E ¯1 ∪ · · · ∪ E ¯k ) P(E) = 1 − P(E) 1−

k  i=1

and the result follows.

¯i ) = 1 − P(E

k  i=1

Q(xi , y) > 1 − k ·

1 =0 k 

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Notice that if we can find a conjugacy class C = y G with the property that each non-identity element of G generates G with at least 2/3 of the elements in C, then Q(x, y) < 1/3 for all 1 = x ∈ G and thus u(G)  3 by Lemma 3.5. This is the main strategy adopted in the proof of Theorem 3.4. In order to effectively apply Lemma 3.5, we need to be able to estimate the probability Q(x, y). Here the key result is the following lemma (as before, we define M(y) to be the set of maximal subgroups of G containing y). Lemma 3.6 For x, y ∈ G, we have 

Q(x, y) 

fpr(x, G/H).

H∈M(y)

Proof If z ∈ y G then G = x, z if and only if x , y  H for some x ∈ xG and H ∈ M(y). Therefore,  Px (H), Q(x, y)  H∈M(y)

where Px (H) =

|xG ∩ H| = fpr(x, G/H) |xG |

is the probability that a random conjugate of x lies in H. The result follows.



If we can find an element y ∈ G and a positive integer k such that 

fpr(x, G/H)
0 there exists an absolute constant k = k() such that Pk (H) > 1 −  for any maximal subgroup H of an almost simple group. To describe the main steps in the proof, we need some additional notation. Let G be a finite group and set ν(G) = min{k ∈ N : Pk (G)  1/e}. Up to a small multiplicative constant, it is known that ν(G) is the expected number of random elements generating G (see [62] and [50, Proposition 1.1]). If A is a nonabelian chief factor of G, let rkA (G) be the maximal number r such that a normal section of G is the direct product of r chief factors of G isomorphic to A, and let

(A) be the minimal degree of a faithful transitive permutation representation of A. Set

 log rkA (G) δ(G) = max , (6) A log (A) where A runs through the non-abelian chief factors of G. We can now state a remarkable theorem of Jaikin-Zapirain and Pyber [33, Theorem 1.1], which is the key ingredient in the proof of Theorem 4.9. Theorem 4.10 There exist absolute constants α, β ∈ N such that α(d(G) + δ(G)) < ν(G) < βd(G) + δ(G) for any finite group G. Let H be a maximal subgroup of an almost simple group, so d(H)  6 by Theorem 4.3. By considering the structure of H (with the aid of the aforementioned subgroup structure theorems), it is not too difficult to show that H has at most three non-abelian chief factors (see [14, Lemma 8.2]) and thus δ(H) < 1. Therefore, Theorem 4.10 implies that ν(H) < 6β + 1. To complete the proof of Theorem 4.9, set c = 6β + 1 and fix  > 0. Let m be the smallest positive integer such that (1 − 1/e)m <  and set k = cm. Then 1 − Pk (H)  (1 − Pc (H))m  (1 − 1/e)m <  and thus Pk (H) > 1 −  as required.

222 4.3

Burness: Simple groups, generation and probabilistic methods Subgroup growth

Let G be a finite group. We define the depth of a subgroup H of G to be the maximal length of a chain of subgroups from H to G (with proper inclusions). In particular, H is maximal if and only if it has depth 1. We say that a subgroup of depth 2 is a second maximal subgroup of G, and so on. It will be convenient to introduce the following notation: Mk (G) = {H : H  G has depth k}, mk,n (G) = |{H ∈ Mk (G) : [G : H] = n}|. For example, m1,n (G) is the number of maximal subgroup of G with index n. For a fixed value of k, it is interesting to consider the growth of mk,n (G) as a function of n. For example, if G is an infinite family of finite groups then we can ask if there is an absolute constant c such that m1,n (G) < nc for all n and all G ∈ G. If this condition holds, then we say that the groups in G have polynomial maximal subgroup growth. For example, if p is a prime and G = (Zp )d then m1,p (G) =

pd − 1 ∼ pd−1 , p−1

so elementary abelian p-groups do not have this property. This growth condition arises naturally in the study of profinite groups. Recall that a profinite group G is positively finitely generated (PFG) if Pk (G) > 0 for some positive integer k, where Pk (G) is defined in terms of topological generation if G is infinite. By a celebrated theorem of Mann and Shalev [54], G is PFG if and only if it has polynomial maximal subgroup growth. The next result is a combination of [14, Corollaries 5 and 6]. Theorem 4.11 Almost simple groups have polynomial maximal and second maximal subgroup growth. That is, there exists an absolute constant c such that max{m1,n (G), m2,n (G)} < nc for all almost simple groups G and all n. To prove this, we need the following result, which combines Theorem 4.10 with a result of Lubotzky (see [50, Proposition 1.2]). Theorem 4.12 There exists an absolute constant γ ∈ N such that m1,n (G) < nγd(G)+δ(G) for all finite groups G and all n ∈ N. Let G be an almost simple group. Then d(G)  3 by Theorem 3.11 and it is easy to see that δ(G) = 0 (see (6)), so Theorem 4.12 yields m1,n (G) < n3γ .

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As previously noted, δ(H) < 1 for all H ∈ M1 (G), so m2,n (G) 



m1,a (G) max{m1,n/a (H) : H ∈ M1 (G), [G : H] = a}

a|n


(S2 )p+1 .S4 = H. By applying Lemma 4.1 we conclude that d(H)  p/24 + 1. It would be interesting to seek an appropriate extension of Theorem 4.16 for third maximal subgroups H (and more generally, depth k subgroups) of almost simple groups. References [1] M. Aschbacher, On the maximal subgroups of the finite classical groups, Invent. Math. 76 (1984), 469–514. [2] M. Aschbacher, On intervals in subgroup lattices of finite groups, J. Amer. Math. Soc. 21 (2008), 809–830. [3] M. Aschbacher and R. Guralnick, Some applications of the first cohomology group, J. Algebra 90 (1984), 446–460. [4] G. Binder, The two-element bases of the symmetric group, Izv. Vyss. Ucebn. Zaved. Matematika 90 (1970), 9–11.

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[5] J.D. Bradley and P.E. Holmes, Improved bounds for the spread of sporadic groups, LMS J. Comput. Math. 10 (2007), 132–140. [6] J.D. Bradley and J. Moori, On the exact spread of sporadic simple groups, Comm. Algebra 35 (2007), 2588–2599. [7] J.N. Bray, D.F. Holt and C.M. Roney-Dougal, The Maximal Subgroups of the Lowdimensional Finite Classical Groups, London Math. Soc. Lecture Note Series, vol. 407, Cambridge University Press, 2013. [8] J.L. Brenner and J. Wiegold, Two-generator groups I, Michigan Math. J. 22 (1975), 53–64. [9] T. Breuer, GAP computations concerning Hamiltonian cycles in the generating graphs of finite groups, preprint, 2012 (arxiv:0911.5589). [10] T. Breuer, R.M. Guralnick and W.M. Kantor, Probabilistic generation of finite simple groups, II, J. Algebra 320 (2008), 443–494. [11] T. Breuer, R.M. Guralnick, A. Lucchini, A. Mar´oti, and G.P. Nagy, Hamiltonian cycles in the generating graph of finite groups, Bull. London Math. Soc. 42 (2010), 621–633. [12] T.C. Burness, Fixed point ratios in actions of finite classical groups, I, J. Algebra 309 (2007), 69–79. [13] T.C. Burness and S. Guest On the uniform spread of almost simple linear groups, Nagoya Math. J. 209 (2013), 35–109. [14] T.C. Burness, M.W. Liebeck and A. Shalev, Generation and random generation: from simple groups to maximal subgroups, Adv. Math. 248 (2013), 59–95. [15] T.C. Burness, M.W. Liebeck and A. Shalev, Generation of second maximal subgroups and the existence of special primes, Forum Math. Sigma 5 (2017), e25, 41 pp. [16] P.J. Cameron, A. Lucchini and C.M. Roney-Dougal, Generating sets of finite groups, Trans. Amer. Math. Soc. 370 (2018), 6751–6770. [17] M.D.E. Conder, Generators for alternating and symmetric groups, J. London Math. Soc. 22 (1980), 75–86. [18] M.D.E. Conder, An update on Hurwitz groups, Groups Complex. Cryptol. 2 (2010), 35–49. [19] J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker and R.A. Wilson, Atlas of Finite Groups, Oxford University Press, 1985. [20] F. Dalla Volta and A. Lucchini, Generation of almost simple groups, J. Algebra 178 (1995), 194–223. [21] J.D. Dixon, The probability of generating the symmetric group, Math. Z. 110 (1969), 199-205. [22] J.D. Dixon and B. Mortimer, Permutation groups, Graduate Texts in Math., vol. 163, Springer-Verlag, New York, 1996. [23] F. Erdem, On the generating graphs of symmetric groups, J. Group Theory 21 (2018), 629–649. [24] P. Erd˝os and P. Tur´ an, On some problems of a statistical group-theory, II, Acta. Math. Acad. Sci. Hung. 18 (1967), 151–163. [25] B. Everitt, Alternating quotients of Fuchsian groups, J. Algebra 223 (2000), 457–476. [26] B. Fairbairn, The exact spread of M23 is 8064, Int. J. Group Theory 1 (2012), 1–2. [27] B. Fairbairn, New upper bounds on the spreads of the sporadic simple groups, Comm. Algebra 40 (2012), 1872–1877. [28] S. Guest, J. Morris, C.E. Praeger and P. Spiga, On the maximum orders of elements of finite almost simple groups and primitive permutation groups, Trans. Amer. Math. Soc. 367 (2015), 7665–7694. [29] R.M. Guralnick and W.M. Kantor, Probabilistic generation of finite simple groups, J. Algebra 234 (2000), 743–792.

228

Burness: Simple groups, generation and probabilistic methods

[30] R. Guralnick, T. Pentilla, C.E. Praeger, and J. Saxl, Linear groups with orders having certain large prime divisors, Proc. London Math. Soc. 78 (1999), 167–214. [31] R.M. Guralnick and A. Shalev, On the spread of finite simple groups, Combinatorica 23 (2003), 73–87. [32] S. Harper, On the uniform spread of almost simple symplectic and orthogonal groups, J. Algebra 490 (2017), 330–371. [33] A. Jaikin-Zapirain and L. Pyber, Random generation of finite and profinite groups and group enumeration, Annals of Math. 173 (2011), 769–814. [34] S. Jambor, A. Litterick and C. Marion, On finite simple images of triangle groups, Israel J. Math. 227 (2018), 131–162. [35] W.M. Kantor and A. Lubotzky, The probability of generating a finite classical group, Geom. Dedicata 36 (1990), 67–87. [36] C.S.H. King, Generation of finite simple groups by an involution and an element of prime order, J. Algebra 478 (2017), 153–173. [37] P.B. Kleidman and M.W. Liebeck, The Subgroup Structure of the Finite Classical Groups, London Math. Soc. Lecture Note Series, vol. 129, Cambridge University Press, 1990. [38] M. Larsen, A. Lubotzky and C. Marion, Deformation theory and finite simple quotients of triangle groups I, J. Eur. Math. Soc. (JEMS) 16 (2014), 1349–1375. [39] R. Lawther, M.W. Liebeck and G.M. Seitz, Fixed point ratios in actions of finite exceptional groups of Lie type, Pacific J. Math. 205 (2002), 393–464. [40] M.W. Liebeck, Probabilistic and asymptotic aspects of finite simple groups, in Probabilistic group theory, combinatorics, and computing, 1–34, Lecture Notes in Math., 2070, Springer, London, 2013. [41] M.W. Liebeck and J. Saxl, Minimal degrees of primitive permutation groups, with an application to monodromy groups of coverings of Riemann surfaces, Proc. London Math. Soc. 63 (1991), 266–314. [42] M.W. Liebeck and G.M. Seitz, A survey of of maximal subgroups of exceptional groups of Lie type, in Groups, combinatorics & geometry (Durham, 2001), 139–146, World Sci. Publ., River Edge, NJ, 2003. [43] M.W. Liebeck and A. Shalev, The probability of generating a finite simple group, Geom. Dedicata 56 (1995), 103–113. [44] M.W. Liebeck and A. Shalev, Simple groups, probabilistic methods, and a conjecture of Kantor and Lubotzky, J. Algebra 184 (1996), 31–57. [45] M.W. Liebeck and A. Shalev, Classical groups, probabilistic methods, and the (2, 3)generation problem, Annals of Math. 144 (1996), 77–125. [46] M.W. Liebeck and A. Shalev, Random (r, s)-generation of finite classical groups, Bull. London Math. Soc. 34 (2002), 185–188. [47] M.W. Liebeck and A. Shalev, Fuchsian groups, coverings of Riemann surfaces, subgroup growth, random quotients and random walks, J. Algebra 276 (2004), 552–601. [48] M.W. Liebeck and A. Shalev, Fuchsian groups, finite simple groups and representation varieties, Invent. Math. 159 (2005), 317–367. [49] F. L¨ ubeck and G. Malle, (2, 3)-generation of exceptional groups, J. London Math. Soc. 59 (1999), 109–122. [50] A. Lubotzky, The expected number of random elements to generate a finite group, J. Algebra 257 (2002), 452–459. [51] A. Lucchini and F. Menegazzo, Generators for finite groups with a unique minimal normal subgroup, Rend. Semin. Mat. Univ. Padova 98 (1997), 173–191. [52] A.M. Macbeath, Generators of the linear fractional groups, Proc. Sympos. Pure Math., vol. XII (Amer. Math. Soc., 1969), pp.14–32. [53] G. Malle, J. Saxl and T. Weigel, Generation of classical groups, Geom. Dedicata 49

Burness: Simple groups, generation and probabilistic methods

229

(1994), 85–116. [54] A. Mann and A. Shalev, Simple groups, maximal subgroups, and probabilistic aspects of profinite groups, Israel J. Math. 96 (1996), 449–468. [55] C. Marion, Triangle groups and PSL2 (q), J. Group Theory 12 (2009), 689–708. [56] C. Marion, On triangle generation of finite groups of Lie type, J. Group Theory 13 (2010), 619–648. [57] A. McIver and P. Neumann, Enumerating finite groups, Quart. J. Math. Oxford 38 (1987), 473–488. [58] N.E. Menezes, M. Quick and C.M. Roney-Dougal, The probability of generating a finite simple group, Israel J. Math. 198 (2013), 371–392. [59] G.A. Miller, On the groups generated by two operators, Bull. Amer. Math. Soc. 7 (1901), 424–426. [60] L. Morgan and C.M. Roney-Dougal, A note on the probability of generating alternating or symmetric groups, Arch. Math. (Basel) 105 (2015), 201–204. [61] E. Netto, Substitutionentheorie und ihre Anwendungen auf die Algebra, Teubner, Leipzig, 1882; English transl. 1892, second edition, Chelsea, New York, 1964. [62] I. Pak, On probability of generating a finite group, preprint, 1999. [63] P.P. P´ alfy, On Feit’s examples of intervals in subgroup lattices, J. Algebra 116 (1988), 471–479. [64] M.A. Pellegrini, The (2, 3)-generation of the special linear groups over finite fields, Bull. Aust. Math. Soc. 95 (2017), 48–53. [65] M.A. Pellegrini and M.C. Tamburini, Finite simple groups of low rank: Hurwitz generation and (2, 3)-generation, Int. J. Group Theory 4 (2015), 13–19. [66] S. Piccard, Sur les bases du groupe sym´etrique et du groupe alternant, Math. Ann. 116 (1939), 752–767. [67] L.L. Scott, Matrices and cohomology, Annals of Math. 105 (1977), 473–492. [68] A. Shalev, Probabilistic group theory and Fuchsian groups, in Infinite groups: geometric, combinatorial and dynamical aspects, 363–388, Progr. Math., 248, Birkh¨ auser, Basel, 2005. [69] A. Stein, 1 12 -generation of finite simple groups, Beitr¨ age Algebra Geom. 39 (1998), 349–358. [70] R. Steinberg, Generators for simple groups, Canad. J. Math. 14 (1962), 277–283. [71] M. Suzuki, On a class of doubly transitive groups, Annals of Math. 75 (1962), 105– 145. [72] T.S. Weigel, Generation of exceptional groups of Lie-type, Geom. Dedicata 41 (1992), 63–87. [73] H. Wielandt, Finite Permutation Groups, Academic Press, New York (1964). [74] A.J. Woldar, On Hurwitz generation and genus actions of sporadic groups, Illinois Math. J. 33 (1989), 416–437. [75] K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. Math. Phys. 3 (1892), 265– 284.

IRREDUCIBLE SUBGROUPS OF SIMPLE ALGEBRAIC GROUPS – A SURVEY TIMOTHY C. BURNESS∗ and DONNA M. TESTERMAN† ∗

School of Mathematics, University of Bristol, Bristol BS8 1TW, UK Email: [email protected] † ´ Institut de Math´ematiques, Station 8, Ecole Polytechnique F´ed´erale de Lausanne, CH-1015 Lausanne, Switzerland Email: donna.testerman@epfl.ch

Abstract Let G be a simple linear algebraic group over an algebraically closed field K of characteristic p  0, let H be a proper closed subgroup of G and let V be a nontrivial finite dimensional irreducible rational KG-module. We say that (G, H, V ) is an irreducible triple if V is irreducible as a KH-module. Determining these triples is a fundamental problem in the representation theory of algebraic groups, which arises naturally in the study of the subgroup structure of classical groups. In the 1980s, Seitz and Testerman extended earlier work of Dynkin on connected subgroups in characteristic zero to all algebraically closed fields. In this article we will survey recent advances towards a classification of irreducible triples for all positive dimensional subgroups of simple algebraic groups.

1

Introduction

Let G be a group, let H be a proper subgroup of G and let V be a nontrivial finite dimensional irreducible KG-module for some field K. Let us write V |H to denote the restriction of V to H. Then (G, H, V ) is an irreducible triple if and only if V |H is irreducible. In this survey article we study irreducible triples in the special case where G is a simple linear algebraic group over an algebraically closed field K of characteristic p  0, H is positive dimensional and closed, and V is a rational KG-module. Throughout this article all algebraic groups considered are linear; henceforth, we use the term “algebraic group” in place of “linear algebraic group”. The study of these triples was initiated by Dynkin [9, 10] in the 1950s (for K = C) and it was subsequently extended by Seitz [23] (classical groups) and Testerman [26] (exceptional groups) in the 1980s to arbitrary algebraically closed fields. The latter work of Seitz and Testerman provides a classification of the irreducible triples (G, H, V ) in the case where H is connected, with the extra condition that if G is classical then V is not the natural module, nor its dual (this assumption is unavoidable). More recently, several authors have focussed on extending this classification to all positive dimensional subgroups and this goal has now essentially been achieved in the series of papers [4, 5, 6, 11, 12, 15]. In this article, we will describe some of these recent advances, highlighting the main ideas and techniques.

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As we will discuss in Section 4.2, it has very recently come to light that there is an omission in the statement of the main theorem of [23]. More specifically, there is a gap in the proof of [23, (8.7)] and a new infinite family of irreducible triples has been discovered by Cavallin and Testerman [7] (together with an additional missing case for (G, H, p) = (C4 , B3 , 2)). With a view towards future applications, one of the goals of this article is to record an amended version of Seitz’s theorem from [23]. In addition, we will show that the new family of triples does not have any effect on the main results in [4, 5] on irreducible triples for disconnected maximal subgroups (see Section 5 for the details). Irreducible triples arise naturally in the study of the subgroup structure of classical algebraic groups, which has been an area of intensive research for several decades. To illustrate the connection, let us consider the group G = SL(V ), where V is a finite dimensional vector space over an algebraically closed field K, and the problem of classifying the maximal closed positive dimensional subgroups of G. Let H be a closed positive dimensional subgroup of G. As a special case of an important reduction theorem of Liebeck and Seitz [18, Theorem 1] (see Theorem 3.1), one of the following holds: (i) H is contained in a subgroup M , where M is the stabilizer in G of a proper non-zero subspace of V , or the stabilizer of a nontrivial direct sum or tensor product decomposition of V , or the stabilizer of a nondegenerate form on V . (ii) The connected component H ◦ is simple and acts irreducibly and tensor indecomposably on V , and does not preserve a nondegenerate form on V . It is straightforward to determine which of the subgroups M as in (i) are maximal among closed subgroups; indeed with very few exceptions, these subgroups are maximal in G. However, it is not so easy to determine when subgroups H satisfying (ii) are maximal. If such a subgroup H is not maximal, then it must be properly contained in a closed subgroup L < G. Moreover, L◦ must itself be a simple algebraic group acting irreducibly (and tensor indecomposably) on V . In addition, if L◦ is a classical group then V is not the natural module (nor its dual) for L◦ . Therefore, (L◦ , H ◦ , V ) is an irreducible triple of the form studied by Seitz and Testerman [23, 26] and we have now “reduced” the determination of the maximal closed positive dimensional subgroups of G to the classification of these irreducible triples. Let us say a few words on the organisation of this article. In Sections 2 and 3 we briefly recall some of the main results and terminology we will need on the representation theory and subgroup structure of simple algebraic groups. In particular, we discuss the parametrization of irreducible modules in terms of highest weights and we present a key theorem of Liebeck and Seitz [18] on the maximal subgroups of classical algebraic groups (see Theorem 3.1). Seitz’s theorem [23] on connected irreducible subgroups is then the main focus of Section 4; we briefly outline some of the main ideas in the proof and we discuss the new examples discovered by Cavallin and Testerman [7]. This allows us to present a corrected version of Seitz’s result (see Theorem 4.1, together with Table 3 in Section 6).

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In Section 5 we present recent work of Burness, Ford, Ghandour, Marion and Testerman [4, 5, 11, 12], which has extended Seitz’s theorem to disconnected positive dimensional maximal subgroups of classical algebraic groups. Here the analysis naturally falls into two cases according to the type of maximal subgroup (in terms of the methods, there is an important distinction between geometric and nongeometric maximal subgroups). The main result is Theorem 5.4, with the relevant irreducible triples presented in Table 4. We present a couple of concrete cases (see Examples 5.7 and 5.11) to illustrate the main ideas in the proof. Since some of the arguments rely on Seitz’s main theorem for connected subgroups, we also take the opportunity to verify the statements of the main theorems in [4, 5] in light of the new examples discovered in [7] (see Propositions 5.8 and 5.12). Finally, in Section 6 we present the tables referred to in Theorems 4.1 and 5.4. Acknowledgments. This survey article is based on the content of the second author’s one-hour lecture at the Groups St Andrews conference, which was hosted by the University of Birmingham in August 2017. It is a pleasure to thank the organizers of this meeting for their generous hospitality. The second author acknowledges the support of the Swiss National Science Foundation, grant number 200021-156583. In addition, the material is based upon work supported by the US National Science Foundation under Grant No. DMS-1440140 while the second author was in residence at the Mathematical Sciences Research Institute in Berkeley, California during the Spring 2018 semester. Finally, both authors are very grateful to Mikko Korhonen, Martin Liebeck, Gunter Malle and Gary Seitz for their helpful comments on an earlier draft of this article.

2

Representation theory

Let G be a simply connected simple algebraic group of rank n defined over an algebraically closed field K of characteristic p  0. In this section we introduce some standard notation and terminology, and we briefly recall the parametrization of finite dimensional irreducible KG-modules in terms of highest weights, which will be needed for the statements of the main results in Sections 4 and 5. We refer the reader to [21, Chapters 15 and 16] for further details on the basic theory, and [17] for a more in-depth treatment. Let B = U T be a Borel subgroup of G containing a fixed maximal torus T of G, where U denotes the unipotent radical of B. Let Π(G) = {α1 , . . . , αn } be a corresponding base of the root system Σ(G) = Σ+ (G) ∪ Σ− (G) of G, where Σ+ (G) and Σ− (G) denote the positive and negative roots of G, respectively (throughout this article, we adopt the standard labelling of simple roots and Dynkin diagrams given in Bourbaki [3]). Moreover, we will often identify a simple algebraic group S with its root system, writing S = Σm , where Σm is one of Am , Bm , Cm , Dm , E8 , E7 , E6 , F4 , G2 , to mean that S has root system of the given type, and use the terminology “S is of type Σm ”. In particular, for such S, if J is a simply connected simple group with

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the same root system type as S, then there is a surjective morphism from J to S, and the representation theory of S is controlled by that of J. Let X(T ) ∼ = Zn denote the character group of T and let {λ1 , . . . , λn } be the fundamental dominant weights for T corresponding to our choice of base Π(G), so λi , αj = δi,j for all i and j, where λ, α = 2

(λ, α) , (α, α)

( , ) is an inner product on X(T )R = X(T ) ⊗Z R, and δi,j is the familiar Kronecker delta. In addition, let sα : X(T )R → X(T )R be the reflection relative to α ∈ Σ(G), defined by sα (λ) = λ − λ, α α, and set si = sαi . The finite group W (G) = si | 1  i  n < GL(X(T )R ) is called the Weyl group of G. In addition, we will write Uα = {xα (c) | c ∈ K} to denote the T -root subgroup of G corresponding to α ∈ Σ(G). Finally, if H is a closed reductive subgroup of G and TH ◦ is a maximal torus of H ◦ contained in T then we will sometimes write λ|H ◦ to denote the restriction of λ ∈ X(T ) to the subtorus TH ◦ . Let V be a finite dimensional KG-module. The action of T on V can be diagonalized, giving a decomposition 1 V = Vμ , μ∈X(T )

where Vμ = {v ∈ V | tv = μ(t)v for all t ∈ T }. A character μ ∈ X(T ) with Vμ = 0 is called a weight (or T -weight) of V , and Vμ is its corresponding weight space. The dimension of Vμ , denoted by mV (μ), is called the multiplicity of μ. We write Λ(V ) for the set of weights of V . For any weight μ ∈ Λ(V ) and any root α ∈ Σ(G) we have  Uα v ⊆ v + Vμ+mα (1) m∈N+

for all v ∈ Vμ . There is a natural action of the Weyl group W (G) on X(T ), which in turn induces an action on Λ(V ). In particular, Λ(V ) is a union of W (G)-orbits, and all weights in a W (G)-orbit have the same multiplicity. By the Lie-Kolchin theorem, the Borel subgroup B stabilizes a 1-dimensional subspace v + of V ; the action of B on v + affords a homomorphism χ : B → K ∗ with kernel U . Therefore χ can be identified with a character λ ∈ X(T ), which is a weight of V . If V is an irreducible KG-module then the 1-space v + is unique, V = Gv + , mV (λ) = 1 and each weight μ ∈ Λ(V ) is obtained from λ by subtracting some positive roots. Consequently, we say that λ is the highest weight of V , and v + is a maximal vector. Since G is simply connected, the fundamental dominant weights form a Z-basis for the additive group of all weights for T , and a weight λ is said to be dominant

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 if λ = ni=1 ai λi and each ai is a non-negative integer. If V is a finite dimensional irreducible KG-module then its highest weight is dominant. Conversely, given any dominant weight λ one can construct a finite dimensional irreducible KGmodule with highest weight λ. Moreover, this correspondence defines a bijection between the set of dominant weights of G and the set of isomorphism  classes of finite dimensional irreducible KG-modules. For a dominant weight λ = ni=1 ai λi we write LG (λ) for the irreducible KG-module with highest weight λ. If char(K) = p > 0, we say that LG (λ) is p-restricted (or that λ is a p-restricted weight) if ai < p for all i. It will be convenient to say that every dominant weight is p-restricted when p = 0. We will often represent the KG-module LG (λ) by an appropriately labelled Dynkin diagram (see Tables 3 and 4, for example). For instance, 1

1

denotes the irreducible KG-module LG (λ2 + λ4 ) for G = A5 = SL6 (K). In characteristic zero, the dimension of LG (λ) is given by Weyl’s dimension formula  (α, λ + ρ) dim LG (λ) = , (α, ρ) + 1

α∈Σ (G)

where ρ = 2 α∈Σ+ (G) α. However, there is no known formula for dim LG (λ) when p > 0 (the above expression is an upper bound), apart from some special cases (see [22] and [5, Table 2.1], for example). Using computational methods, L¨ ubeck [20] has calculated the dimension of every irreducible KG-module in all characteristics, up to some fixed dimension depending on the type of G. For instance, the irreducible modules for SL3 (K) of dimension at most 400 can be read off from [20, Table A.6]. In positive characteristic p > 0, the Frobenius automorphism of K, Fp : K → K, c → cp , induces an endomorphism F : G → G of algebraic groups defined by xα (c) → xα (cp ), for all α ∈ Σ(G), c ∈ K. Given a rational KG-module V and corresponding representation ϕ : G → GL(V ), we can use F to define a new i representation ϕ(p ) : G → GL(V ) for each integer i  1. The corresponding KGi i module is denoted by V (p ) and the action is given by ϕ(p ) (g)v = ϕ(F i (g))v for i) (p g ∈ G, v ∈ V . We say that V is a Frobenius twist of V . By Steinberg’s tensor product theorem (see [21, Theorem 16.12], for example), every finite dimensional irreducible KG-module V is a tensor product of Frobenius twists of p-restricted KG-modules, so it is natural to focus on the p-restricted modules. Indeed, if H is a subgroup of G and V = V1 ⊗ V2 is a tensor product of KG-modules, then V is irreducible as a KH-module only if V1 and V2 are both irreducible KH-modules. This observation essentially reduces the problem of determining the irreducible triples for G to the p-restricted case, whence the main results in Sections 4 and 5 will be stated in terms of p-restricted tensor indecomposable KG-modules. Finally, we record the following technical lemma, which will be relevant later. Lemma 2.1 Let G be a simply connected simple algebraic group of type Bn or Cn with n  3, defined over an algebraically closed field of characteristic 2. Let λ be

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a 2-restricted dominant weight for G and let ϕ : G → GL(V ) be the associated irreducible representation. Then ϕ(G) is again of type Bn , respectively Cn , if and only if λ, α =  0 for some short simple root α of G. Proof Let dϕ : Lie(G) → gl(V ) be the differential of ϕ, where Lie(G) is the Lie algebra of G. By a theorem of Curtis [8], dϕ is irreducible and thus dϕ = 0. Recall that Lie(G) has a proper ideal containing all short root vectors. This ideal lies in ker(dϕ) if and only if λ, α = 0 for all short simple roots α of G. Moreover, im(dϕ) is an ideal of Lie(ϕ(G)) and the result now follows by inspecting the ideal structure of Lie(G) (see [16], for example). 

3

Subgroup structure

Let G be a simple algebraic group over an algebraically closed field K of characteristic p  0. By Chevalley’s classification theorem for semisimple algebraic groups, the isomorphism type of G is determined by its root datum (see [21, Section 9.2]) and thus G is either of classical type or exceptional type. In this article, we will focus on irreducible triples for classical type algebraic groups, in which case it will be useful to have a more geometric description of these groups in order to describe their positive dimensional maximal subgroups. Let W be a finite dimensional vector space over K, equipped with a form β, which is either the zero form, or a symplectic or nondegenerate quadratic form (in the latter case, nondegenerate means that the radical of the underlying bilinear form is trivial, unless p = 2 and dim W is odd, in which case the radical is a nonsingular 1-space). Assume dim W  2, and further assume that dim W  3 and dim W = 4 if β is a quadratic form. Let Isom(W, β) be the corresponding isometry group, comprising the invertible linear maps on W which preserve β, and let G = Isom(W, β) be the derived subgroup. Then G is a simple classical algebraic group over K, namely one of SL(W ), Sp(W ) or SO(W ) (we will sometimes use the notation G = Cl(W ) to denote any one of these groups). Note that Cl(W ) = Isom(W, β) ∩ SL(W ), unless p = 2, in which case SO(W ) has index 2 in GO(W ) ∩ SL(W ). As discussed in the previous section, we will often adopt the Lie notation for groups with the same root system type, so we will write An , Bn , Cn , and Dn to denote a simple algebraic group of classical type. The main theorem on the subgroup structure of simple classical algebraic groups is due to Liebeck and Seitz [18], which can be viewed as an algebraic group analogue of Aschbacher’s celebrated subgroup structure theorem [1] for finite classical groups. In order to state this result, we need to introduce some additional notation. Following [18, Section 1], one defines six natural, or geometric, collections of closed 0 subgroups of G, labelled C1 , . . . , C6 , and we set C = i Ci . A rough description of the subgroups in each Ci collection is given in Table 1 (note that the subgroups comprising the collection C5 are finite). The following result is [18, Theorem 1]. Theorem 3.1 Let H be a closed subgroup of G = Cl(W ). Then either (i) H is contained in a member of C; or

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Table 1. The Ci collections C1 C2 C3 C4 C5 C6

Rough description Stabilizers of subspaces of W + Stabilizers of orthogonal decompositions W = i Wi , dim Wi = a Stabilizers of totally singular decompositions W = W1 ⊕ W2 Stabilizers of tensor product decompositions W = W 21 ⊗ W2 Stabilizers of tensor product decompositions W = i Wi , dim Wi = a Normalizers of symplectic-type r-groups, r = p prime Classical subgroups

(ii) modulo scalars, H is almost simple and E(H) (the unique quasisimple normal subgroup of H) is irreducible on W . Further, if G = SL(W ) then E(H) does not fix a nondegenerate form on W . In addition, if H is infinite then E(H) = H ◦ and W is a tensor indecomposable KH ◦ -module. We refer the reader to [18] and [5, Section 2.5] for further details on the structure of the geometric subgroups comprising C. We will write S to denote the collection of closed subgroups of G that arise in part (ii) of Theorem 3.1 (we sometimes use the term non-geometric to refer to the subgroups in S). Note that it is not feasible to determine the members of S, in general. Indeed, as we remarked in the previous section, we do not even know the dimensions of the irreducible modules for a given simple algebraic group. Example 3.2 If G = Sp(W ) = Sp8 (K) is a symplectic group, then the positive dimensional subgroups H ∈ C ∪ S are as follows: C1 : Here H is either a maximal parabolic subgroup P1 , . . . , P4 , where Pi denotes the stabilizer in G of an i-dimensional totally singular subspace of W , or H = Sp6 (K) × Sp2 (K) is the stabilizer of a nondegenerate 2-space. These subgroups are connected. C2 : The collection C2 comprises both the stabilizers of orthogonal decompositions W = W1 ⊥ W2 , where each Wi is a nondegenerate 4-space, in which case H = Sp4 (K)S2 , together with subgroups of the form H = Sp2 (K)S4 , which stabilize an orthogonal decomposition of W into nondegenerate 2-spaces. C3 : Here H = GL4 (K).2 is the stabilizer of a direct sum decomposition W = W1 ⊕ W2 , where W1 and W2 are totally singular 4-spaces. C4 : This collection is empty if p = 2 (see [18, p.429]), so let us assume p = 2. Here the natural module W admits a tensor product decomposition W = W1 ⊗W2 , where W1 is a 2-dimensional symplectic space and W2 is a 4-dimensional orthogonal space. The stabilizer of this decomposition is a (disconnected) central product Sp(W1 ) ⊗ GO(W2 ). In addition, the remaining subgroups in C4 are of the form (Sp2 (K) ⊗ Sp2 (K) ⊗ Sp2 (K)).S3 ,

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stabilizing a decomposition W = W1 ⊗ W2 ⊗ W3 into a tensor product of symplectic 2-spaces. C6 : If p = 2 then H = GO(W ) = SO(W ).2 is the stabilizer of a nondegenerate quadratic form on W (the collection is empty if p = 2). Note that we view H as a geometric subgroup, even though it satisfies the criteria for membership in S (see Remark 5.3). S: Here p = 2, 3, 5, 7 and H = SL(U ) = SL2 (K) is irreducibly embedded in G via the 8-dimensional irreducible KH-module Sym7 (U ), the seventh symmetric power of U . To conclude this section, let us provide a brief sketch of the proof of Theorem 3.1 in the special case where G = SL(W ) and H is positive dimensional.

Sketch proof of Theorem 3.1 Let us assume G = SL(W ) and H is positive dimensional. First observe that if H fixes a nondegenerate form on W , then H lies in a classical subgroup of G belonging to the collection C6 . Therefore, we may assume otherwise. If H acts reducibly on W , then H lies in the stabilizer of a proper non-zero subspace of W , which is a parabolic subgroup in the collection C1 . Now assume H acts irreducibly on W and consider the action of the connected component H ◦ on W . Since H stabilizes the socle of W |H ◦ , it follows that W |H ◦ is completely reducible. Moreover, the irreducibility of W |H implies that H transitively permutes the homogeneous components of W |H ◦ and thus H stabilizes a direct sum decomposition of the form W = W1 ⊕ · · · ⊕ Wt , where each Wi is mdimensional for some m. In particular, H  (GLm (K)  St ) ∩ G. If W |H ◦ has at least two homogeneous components (that is, if t  2), then H is contained in a C2 -subgroup of G. To complete the argument, we may assume W |H ◦ is homogeneous. If W |H ◦ is reducible then W |H ◦ = U ⊕ · · · ⊕ U for some irreducible KH ◦ -module U . Here dim U  2 (since H is positive dimensional and Z(G) is finite) and further argument shows that W admits a tensor product decomposition W = W1 ⊗ W2 such that H ◦  GL(W1 ) ⊗ 1 and H  NG (SL(W1 ) ⊗ SL(W2 )). In this situation, we conclude that H is contained in a member of the collection C4 . Finally, let us assume H ◦ acts irreducibly on W . Then H ◦ is necessarily reductive as otherwise H ◦ stabilizes the nonzero set of fixed points of its unipotent radical. We then deduce that H ◦ is in fact semisimple as Z(H ◦ ) must act as scalars on W and is therefore finite. So we now have H ◦ = H1 · · · Hr , a commuting product of simple algebraic groups, and W |H ◦ = W1 ⊗ · · · ⊗ Wr , where Wi is an irreducible KHi -module for each i. If r > 1 then H is contained in the stabilizer of the tensor product decomposition W1 ⊗· · ·⊗Wr , which is in the C4 collection. Finally, suppose r = 1, so H ◦ is simple. If W |H ◦ is tensor decomposable then once again we deduce that H is contained in a C4 -subgroup. Otherwise, H satisfies the conditions in part (ii) of the theorem and thus H is a member of the collection S. 

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Connected subgroups

Let G = Cl(W ) be a simple classical algebraic group over an arbitrary algebraically closed field K of characteristic p  0. In this section we are interested in the irreducible triples of the form (G, H, V ), where H is a closed connected subgroup of G and V is a nontrivial irreducible p-restricted rational KG-module. The main theorem is due to Seitz [23], extending earlier work of Dynkin [9, 10] in characteristic zero, which we recall below. We also report on recent work of Cavallin and Testerman [7], which has identified a gap in the proof of Seitz’s theorem, leading to a new family of irreducible triples. We will explain how the gap arises in [23] and with a view towards future applications, we will present a corrected version of Seitz’s theorem, based on the work in [7]. 4.1

Seitz’s theorem

As described in Section 1, the determination of the maximal closed positive dimensional subgroups of the classical algebraic groups relies on a detailed study of irreducible triples. This problem was first investigated in the pioneering work of Dynkin in the 1950s, in the context of complex semisimple Lie algebras. In [9, 10], Dynkin classifies the triples (g, h, V ), where g is a simple complex Lie algebra, h is a semisimple subalgebra of g and V is an irreducible g-module on which h acts irreducibly. For a classical type Lie algebra g, his classification includes 8 infinite families of natural embeddings h ⊂ g, such as sp2m ⊂ sl2m , and in each case he lists the g-modules on which h acts irreducibly (in terms of highest weights). In most cases, the module is some symmetric or exterior power of the natural module for g. In addition, there are 36 isolated examples (g, h, V ) with g a classical Lie algebra of fixed rank. For example, by embedding the simple Lie algebra h of type E6 in sl27 = sl(W ) via one of its irreducible 27-dimensional representations, he shows that h acts irreducibly on Λi (W ) for i = 2, 3, 4. There are exactly 4 pairs (g, h) in Dynkin’s theorem for g of exceptional type. Dynkin’s classification immediately yields a classification of irreducible triples (G, H, V ), where G is a simple algebraic group defined over C and H is a proper closed connected subgroup. In the 1980s, Seitz [23] (classical groups) and Testerman [26] (exceptional groups) extended Dynkin’s analysis to all simple algebraic groups G over all algebraically closed fields. Here, in the positive characteristic setting, several major difficulties arise that are not present in the work of Dynkin, so a completely different approach is needed. For instance, rational modules need not be completely reducible and we have already mentioned that there is no dimension formula for irreducible modules. Similarly, in positive characteristic, irreducible modules may be tensor decomposable (this is clear from Steinberg’s tensor product theorem) and this significantly complicates the analysis. In order to proceed, it is first useful to observe that if (G, H, V ) is an irreducible triple with H connected then H is semisimple. Seitz’s main technique in [23] is induction. Given a parabolic subgroup PH of H, the Borel-Tits theorem (see [2, Corollary 3.9]) implies the existence of a parabolic subgroup P of G such that PH < P and QH = Ru (PH ) < Ru (P ) = Q. Let LH and L be Levi factors

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of PH and P , respectively. By [23, (2.1)], if V |H is irreducible then LH acts irreducibly on the quotient V /[V, Q], which is an irreducible KL -module (here [V, Q] = qv − v | v ∈ V, q ∈ Q ). Moreover, the highest weight of V /[V, Q] as a KL -module is the restriction of the highest weight of V to an appropriate maximal torus of L (this is a variation of a result of Smith [25]). Thus, Seitz proceeds by induction on the rank of H, treating the base case H = A1 by ad hoc methods, exploiting the fact that all weights of an irreducible KA1 -module have multiplicity one. In the main theorem of [23] (which includes Testerman’s results from [26] for exceptional groups), the embedding of H in G is described in terms of the action of H on the natural KG-module W , in case G is of classical type. For an exceptional group G, an explicit construction of the embedding is required, as given in [26, 27]. Further, the highest weights of V |H and V = LG (λ) are recorded in terms of fundamental dominant weights for H and G (via labelled Dynkin diagrams). Comparing the main theorems of [9, 10] and [23], we observe that Dynkin’s classification of triples (for K = C) is a subset of the list of irreducible triples that arise in positive characteristic p > 0, with some additional conditions depending on p. For example, consider the natural embedding of H = Sp(W ) in G = SL(W ). In characteristic 0, H acts irreducibly on the symmetric power Symi (W ) for all i  0, whereas in positive characteristic we need the condition 0  i < p. In addition, there are infinite families of triples that only arise in positive characteristic. For instance, if we take H < G as above and assume p > 2, then H also acts irreducibly on the KG-modules with highest weight λ = aλk + bλk+1 , where 1k
· · · > 0,

(4)

where [W, Q0X ] = W and [W, QiX ] = qw − w : w ∈ [W, Qi−1 X ], q ∈ QX for i  1. Set Wi = [W, QiX ]/[W, Qi+1 X ] for each i  0 and let P = QL be the stabilizer in G of this flag, which is a parabolic subgroup of G with unipotent radical

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Q and Levi factor L, with several desirable properties. In particular, QX < Q (see [4, Lemma 2.7.1]). We can study the weights occurring in [W, QiX ] in order to obtain a lower bound on dim Wi , which then leads to structural information on L . Now a theorem of Smith [25] implies that L acts irreducibly on the quotient V /[V, Q], and similarly LX on each Vi /[Vi , QX ]. By combining this observation with our lower bounds on the dimensions of the quotients in (4), we can obtain restrictions on the highest weight λ of V . For instance, if we consider the case where PX is a t-stable Borel subgroup of X, then [11, Lemma 5.1] implies that L has an A1 factor and this severely restricts the coefficients of λ corresponding to the roots in L (this is illustrated in Example 5.11 below). Beyond the choice of a t-stable Borel subgroup, we often study the embedding of other parabolic subgroups PX of X in parabolic subgroups of G, again with an inclusion of unipotent radicals as above. In the general setting, both L and LX act on the space V /[V, Q], and as a module for LX we have V /[V, Q] = V1 /[V1 , QX ] ⊕ V2 /[V2 , QX ] or Vi /[Vi , QX ] for i = 1, 2. If we write L = L1 · · · Lk , a commuting product of simple algebraic groups, then V /[V, Q] = M1 ⊗ · · · ⊗ Mk , where each Mi is an irreducible KLi -module. In particular, if Mi is nontrivial then we may consider the triple (Li , πi (LX ), Mi ), where πi : LX → Li is the natural projection map induced by the isomorphisms and inclusions LX ∼ = PX /QX ∼ = PX Q/Q < P/Q ∼ = L. It follows that (Li , πi (LX ), Mi ) is either an irreducible triple, or LX has precisely two composition factors on Mi . This allows us to apply induction on the rank of G, using the previously considered smaller rank cases to obtain information on the highest weights of the Mi , and ultimately to determine the highest weight λ of V . Example 5.11 Suppose H = A2 t = A2 .2 and p = 2, 5, so X = A2 and t is an involutory graph automorphism of X. Let {δ1 , δ2 } be the fundamental dominant weights for X corresponding to the simple roots {β1 , β2 }. Set W = LX (δ), where δ = 2δ1 + 2δ2 , so dim W = 27 and the action of X on W embeds X in G = SO(W ). Moreover, the symmetry of δ implies thatW is self-dual, so t also acts on W and thus H < G. Set V = LG (λ), where λ = i ai λi is p-restricted and λ = λ1 . Claim.

V |H is reducible.

Seeking a contradiction, let us assume V |H is irreducible. By the main theorem of [23], V |X is reducible and so Clifford theory implies that V |X = V1 ⊕V2 , where Vi = LX (μi ). Without loss of generality, we may assume that μ1 = λ|X = c1 δ1 + c2 δ2 , in which case μ2 is the image of μ1 under the action of t, so μ2 = c2 δ1 + c1 δ2 . Moreover, the irreducibility of V |H implies that V1 and V2 are non-isomorphic (see [4, Proposition 2.6.2]), whence c1 = c2 .

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Let PX = QX LX be a t-stable Borel subgroup of X (so LX = TX is a maximal torus of X) and consider the corresponding flag of W given in (4). Define the quotients Wi as above. Using [23, (2.3)], one checks that dim W0 = 1, dim W1 = 2, dim W2 = 4, dim W3 = 4, dim W4 = 5, which implies that L = A1 A3 A3 B2 . By the aforementioned theorem of Smith [25], V /[V, Q] = V1 /[V1 , QX ] ⊕ V2 /[V2 , QX ] = M1 ⊗ M2 ⊗ M3 ⊗ M4 is an irreducible KL -module, with highest weight λ|L (here the notation indicates that Mi is an irreducible module for the i-th factor of L ). But V /[V, Q] is 2dimensional, so M1 must be the natural module for the A1 factor of L , and M2 , M3 and M4 are trivial (see [11, Lemma 5.2]). This observation yields useful information on the coefficients in λ; indeed, we obtain the following partially labelled diagram: a1

1

a3

0

0

0

a7

0

0

0

a11

0

0

In other words, we have reduced to the case where λ = a1 λ1 + λ2 + a3 λ3 + a7 λ7 + a11 λ11 .

(5)

By choosing an appropriate ordering on the TX -weights in W , we can determine the restrictions of the simple roots αi to the subtorus TX (see [4, Table 3.2]). For example, we can fix an ordering so that α2 |X = β2 − β1 . Now, since the coefficient of λ2 in (5) is positive, it follows that λ − α2 is a T -weight of V and thus (λ − α2 )|X = μ1 − β2 + β1 is a TX -weight of V |X . Moreover, by considering root restrictions we can show that this is the highest weight of a KX-composition factor of V , so we must have μ2 = μ1 − β2 + β1 . By [23, (8.6)], δ−β1 −β2 has multiplicity 2 as a TX -weight of W (here we are using the fact that p = 5), which means that we may assume (λ− 6i=1 αi )|X = δ−β1 −β2 and thus α6 |X = 0. It follows that there are at least three T -weights of V which give ν = λ|X − β2 on restriction to TX , so the multiplicity of ν is at least 3. However, one checks that ν has multiplicity 1 in both V1 and V2 , so this is incompatible with the decomposition V |X = V1 ⊕ V2 . This contradiction completes the proof of the claim. Finally, let us address the veracity of the main theorem of [4], in view of the corrections to Seitz’s main theorem discussed in Section 4.2. Proposition 5.12 Let G, H, V be given as in Hypothesis () and let us assume H is non-geometric. Then V |H is irreducible if and only if (G, H, V ) is as given in Theorem 5.9. Proof As discussed above, in the proof of the main result of [4] we proceed by induction on the rank of G, studying the embedding of various parabolic subgroups of H in parabolic subgroups of G. We then rely upon an inductive list of examples,

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for disconnected as well as connected subgroups. In view of the recently discovered configurations described in Section 4.2, we must go back through our proof, paying particular attention to all cases where the Levi factor of the chosen parabolic of H ◦ has type B . But for the groups H satisfying the hypotheses of the proposition, we have H ◦ = Am (with m  2), Dm (with m  4) or E6 . In particular, H ◦ does not have a parabolic subgroup whose Levi factor is of type B , so the arguments in [4] are not affected by the new examples. 

6

Tables

In this final section we present Tables 3 and 4, which describe the irreducible triples arising in the statements of Theorems 4.1 and 5.4. Note that Table 3 can be viewed as a corrected version of [23, Table 1]. In the following two remarks, we explain the set-up and notation in each table. Remark 6.1 Let us describe the notation in Table 3. (a) In the first column we give the label for the particular case in that row of the table; our choice of labels is consistent with [23, Table 1]. As explained in Remark 4.2, we have included the following additional cases in Table 3: S10 , S11 , MR1 , MR2 and we have corrected the conditions in the case labelled IV1 . (b) The second column gives the pair (G, H), plus any additional information on char(K) = p and the rank of G, as needed. If no restrictions on p are presented, then it is to be assumed that the only conditions on p are to ensure that the highest weight of V = LG (λ) (as given in the final column of the table) is p-restricted. For example, in case IV1 we have λ = kλn , so this means that p = 0 or p  k + 1. (c) In terms of the second column, the case labelled IV2 requires special attention. Here H is a subgroup of a central product Bk Bn−k , so there are natural projection maps π1 : H → Bk and π2 : H → Bn−k . Seitz’s notation H → Bk Bn−k < Dn+1 indicates that either H projects onto both simple factors, or some factor is of type B2 and the projection of H to this factor is a group of type A1 acting irreducibly on the spin module for the B2 (in the latter situation, note that we also need the condition p  5, as indicated in the table). In addition, as Seitz notes on [23, p.9], it may be necessary for the projections to involve distinct field twists to ensure the irreducibility of V |H . For instance, if A1 → B2 B2 < D5 then we need different field twists on the embedding of A1 in each B2 factor. This notation is also used in cases S6 and MR5 , with the same meaning. (d) In case IV9 we write (†) in the second column to indicate that the given subgroup H acts irreducibly on exactly one of the two spin modules for G (see Remark 4.2(d)).

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(2 to denote a subsystem (e) In the case labelled MR1 , we use the notation A (4 < subgroup of type A2 corresponding to short roots in G2 (and similarly D F4 in case MR2 ). See Remark 4.2(g) for further details. (f) The embedding of H in G is described in the third column of the table, in terms of the action of H on the KG-module W associated with G. In particular, if G is of classical type then W is just the natural module. Similarly, if G has type G2 , F4 , E6 or E7 then W is the minimal module for G of dimension 7 (6 if p = 2), 26 (25 if p = 3), 27 or 56, respectively (note that the case G = E8 does not arise in Table 3). Typically, H is simple and W |H is irreducible, in which case we give the highest weight of W |H in terms of a set of fundamental dominant weights {δ1 , . . . , δm } for H. This notation is suitably modified in cases IV3 and IV5 , where H = H1 H2 is a central product of two simple groups. Note that in cases T1 and T2 , we write 0 for the highest weight of the trivial module. Following [23, Table 1], in some cases we write “usual” in the third column to indicate that the embedding of H in G is via the usual action of H on the natural KG-module. For example, in the case labelled IV1 , H = Bn is embedded in G = Dn+1 as the stabilizer of a nonsingular 1-space. (g) In the fourth and fifth columns, we give the highest weights of V |H and V , respectively, in terms of labelled Dynkin diagrams, together with any additional conditions on the relevant parameters. Once again, this is consistent with [23, Table 1], with the exception that the conditions in case IV1 have been corrected in view of [7] (see Section 4.2).

Table 3. The irreducible triples with H connected no.

H 2 and H ≤ SL(n, Z) be arithmetic. Given u, v ∈ Qn , the procedure Orbit(u, v) tests whether there is g ∈ SL(n, Z) such that gu = v, and computes g if it exists. Stabilizer(H, u) returns the (finitely generated) stabilizer of u in H. Both procedures solve the related orbit and stabilizer problems for the congruence image over ZM and for the maximal PCS in H acting on Qn . The outputs are then combined. See [12, Section 4]. 3.7

Experiments

The algorithms of this section are joint work with Alexander Hulpke. Below we review some experiments illustrating our GAP implementation of the algorithms and their practicality; see [13, 15, 16] for more. 3.7.1 Integral representations of the fundamental group x, y, z | zxz −1 = xy, zyz −1 = yxy of the figure-eight knot complement are constructed in [26]. For non-zero T ∈ Z, let βT (x) = XT and βT (y) = YT where ⎡

−1 + T 3 −T 0 −1 XT = ⎣ −T 0

⎤ T2 2T ⎦ , 1

⎤ −1 0 0 2 YT = ⎣−T 1 −T ⎦ . T 0 −1 ⎡

Then βT is a homomorphism and βT (x, y ) ≤ SL(3, Z) is arithmetic. Construction of these representations was motivated by long-standing problems; such as the conjecture that each arithmetic group in SL(n, Z) has a 2-generator finite index subgroup. The conjecture has been settled affirmatively [28]. Still, the subgroups XT , YT merit closer scrutiny. Earlier attempts to compute |SL(3, Z) : XT , YT | were stymied by the fact that this index may be arbitrarily large. We were able to compute indices using our algorithms (see [13, Section 4.1]). For example, let T = 100; the index 242 35 525 74 13·312 67·1783 and level 27 56 29·67·193 were found in 892.6 seconds.

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3.7.2 A second family of test groups comes from applications in theoretical physics. Let G(d, k) = U, T where ⎡

1 1 0 ⎢0 1 0 U =⎢ ⎣d d 1 0 −k −1

⎤ 0 0⎥ ⎥, 0⎦ 1

⎡

1 ⎢0 T =⎢ ⎣0 0

0 1 0 0

0 0 1 0

⎤ 0 1⎥ ⎥. 0⎦ 1

For fourteen pairs d, k of integers, G(d, k) ≤ Sp(4, Z) is the monodromy group of a generalized hypergeometric ordinary differential equation associated to Calabi-Yau threefolds. Seven of these groups are arithmetic, while the rest are thin [32, 33]. To investigate the latter, one could attempt to construct arithmetic groups in Sp(4, Z) containing them [6]. We successfully computed cl(G(d, k)) for the seven thin groups [13, Table 3]; e.g., it took 25 seconds to find the level 25 32 and the index 217 36 52 of G(12, 7).

4

Where to next?

We outline avenues for future research. New methods and algorithms for algebraic groups and Lie algebras would have an impact on computing with virtually solvable groups. Despite significant progress (cf. Section 2), key algorithmic questions are still unresolved. One of these is membership testing. This problem is known to be decidable for groups of finite rank. The main challenge is handling the unipotent radical, which is a torsionfree nilpotent group that may not be finitely generated. Lie algebra methods due to P. Hall, and computing in ambient solvable algebraic groups, are possible approaches. These are similarly promising in the design of algorithms for structural analysis of virtually solvable linear groups. We also expect a number of new algorithms for computing with (virtually) nilpotent and (virtually) polycyclic linear groups. Methods based on algebraic group techniques will be productive in applications to non-virtually solvable groups (cf. Section 3). Arithmeticity testing is open in general, even for subgroups of SL(n, Z). Indeed, it is not known whether the problem is decidable. Computing generating sets and presentations of arithmetic subgroups are supplementary problems (cf. [8, Chapter 6], [7]). Construction of free subgroups would aid in the study of matrix groups that are not virtually solvable; ‘large’ free subgroups, i.e., those that are dense in the Zariski closure, are especially useful. Testing freeness of finitely generated linear groups is yet another priority. We await breakthroughs that apply computational methods to the solution of hard problems in group theory, other areas of mathematics, and farther afield (cf. Section 3.7). Here we point to computing linear representations of finitely presented groups: in contrast to the same problem for finite groups, much remains to be done.

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Acknowledgments We are indebted to our collaborators Willem de Graaf, Alexander Hulpke, and Eamonn O’Brien. We also thank Mathematisches Forschungsinstitut Oberwolfach, and the International Centre for Mathematical Sciences, UK, for hosting our visits under their ‘Research in Pairs’ and ‘Research in Groups’ programmes. A. S. Detinko is supported by a Marie Sk lodowska-Curie Individual Fellowship grant (Horizon 2020, EU Framework Programme for Research and Innovation). References [1] R. Aoun, Random subgroups of linear groups are free, Duke Math. J. (1) 160 (2011), 117–173. [2] B. Assmann and B. Eick, Computing polycyclic presentations for polycyclic rational matrix groups, J. Symbolic Comput. (6) 40 (2005), 1269–1284. [3] B. Assmann and B. Eick, Testing polycyclicity of finitely generated rational matrix groups, Math. Comp. 76 (2007), 1669–1682. [4] H. B¨ a¨arnhielm, D. Holt, C. R. Leedham-Green, and E. A. O’Brien, A practical model for computation with matrix groups, J. Symbolic Comput. 68 (2015), 27–60. [5] W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. (3-4) 24 (1997), 235–265. [6] Y. Chen, Y. Yang, and N. Yui, Monodromy of Calabi-Yau differential equations (with an appendix by Cord Erdenberger), J. Reine Angew. Math. 616 (2008), 167–203. [7] R. Coulangeon, G. Nebe, O. Braun, and S. Sch¨ onnenbeck, Computing in arithmetic groups with Voronoi’s algorithm, J. Algebra (1) 435 (2015), 263–285. [8] W. A. de Graaf, Computation with linear algebraic groups (Chapman & Hall/CRC, 2017). [9] H. Derksen, E. Jeandel, and P. Koiran, Quantum automata and algebraic groups, J. Symbolic Comput. (3-4) 39 (2005), 357–371. [10] A. S. Detinko and D. L. Flannery, Algorithms for computing with nilpotent matrix groups over infinite domains, J. Symbolic Comput. (1) 43 (2008), 8–26. [11] A. S. Detinko, D. L. Flannery, and W. A. de Graaf, Integrality and arithmeticity of solvable linear groups, J. Symbolic Comput. 68 (2015), 138–145. [12] A. S. Detinko, D. L. Flannery, and A. Hulpke, Algorithms for arithmetic groups with the congruence subgroup property, J. Algebra 421 (2015), 234–259. [13] A. S. Detinko, D. L. Flannery, and A. Hulpke, Zariski density and computing in arithmetic groups, Math. Comp. 87 (2018), 967–986. [14] A. S. Detinko, D. L. Flannery, and A. Hulpke, GAP functionality for Zariski dense groups, Oberwolfach Preprints, OWP 2017-22. [15] A. S. Detinko, D. L. Flannery, and A. Hulpke, Algorithms for experimenting with Zariski dense subgroups, Exp. Math., to appear. [16] A. S. Detinko, D. L. Flannery, and A. Hulpke, The strong approximation theorem and computing with linear groups, preprint (2018). [17] A. S. Detinko, D. L. Flannery, and E. A. O’Brien, http://magma.maths.usyd.edu. au/magma/handbook/matrix_groups_over_infinite_fields [18] A. S. Detinko, D. L. Flannery, and E. A. O’Brien, Algorithms for the Tits alternative and related problems, J. Algebra 344 (2011), 397–406. [19] A. S. Detinko, D. L. Flannery, and E. A. O’Brien, Algorithms for linear groups of finite rank, J. Algebra 393 (2013), 187–196. [20] A. S. Detinko, D. L. Flannery, and E. A. O’Brien, Recognizing finite matrix groups over infinite fields, J. Symbolic Comput. 50 (2013), 100–109.

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[21] J. Dixon, The structure of linear groups (Van Nostrand Reinhold, London 1971). [22] D. B. A. Epstein, Almost all subgroups of a Lie group are free, J. Algebra 19 (1971), 261–262. [23] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.8.7; 2017, http://www.gap-system.org [24] F. Grunewald and D. Segal, Some general algorithms. I. Arithmetic groups, Ann. of Math. (3) 112 (1980), 531–583. [25] J. C. Lennox and D. J. S. Robinson, The theory of infinite soluble groups (OUP, Oxford 2004). [26] D. D. Long and A. W. Reid, Small subgroups of SL(3, Z), Exp. Math. (4) 20 (2011), 412–425. [27] A. Lubotzky and D. Segal, Subgroup growth, Progress in Mathematics, Vol. 212, (Birkh¨ auser, Basel 2003). [28] C. Meiri, Generating pairs for finite index subgroups of SL(n, Z), J. Algebra 470 (2017), 420–424. [29] E. A. O’Brien, Algorithms for matrix groups, in Groups St Andrews 2009 in Bath. Vol. 2, (C. M. Campbell et al., eds.), London Math. Soc. Lecture Note Ser. 388 (CUP, Cambridge 2011), 297–323. [30] I. Rivin, Large Galois groups with applications to Zariski density, http://arxiv. org/abs/1312.3009v4 [31] P. Sarnak, Notes on thin matrix groups, in Thin groups and superstrong approximation, Math. Sci. Res. Inst. Publ. 61, (CUP, Cambridge 2014), 343–362. [32] S. Singh, Arithmeticity of four hypergeometric monodromy groups associated to Calabi-Yau threefolds, Int. Math. Res. Notices (18), 2015 (2015), 8874–8889. [33] S. Singh and T. Venkataramana, Arithmeticity of certain symplectic hypergeometric groups, Duke Math. J. (3) 163 (2014), 591–617. [34] D. A. Suprunenko, Matrix groups, Translations of Mathematical Monographs, Vol. 45 (AMS, Providence 1976). [35] J. Tits, Free subgroups in linear groups, J. Algebra 20 (1972), 250–270. [36] B. A. F. Wehrfritz, Infinite linear groups (Springer-Verlag, New York 1973). [37] B. A. F. Wehrfritz, Conditions for linear groups to have unipotent derived subgroups, J. Algebra 323 (2010), 3147–3154.

BEAUVILLE p-GROUPS: A SURVEY BEN FAIRBAIRN Department of Economics, Mathematics and Statistics, Birkbeck, University of London, Malet Street, London, WC1E 7HX Email: [email protected]

Abstract Beauville surfaces are a class of complex surfaces defined by letting a finite group G act on a product of Riemann surfaces. These surfaces possess many attractive geometric properties several of which are dictated by properties of the group G. In this survey we discuss the p-groups that may be used in this way. En route we discuss several open problems, questions and conjectures.

1

Introduction

Roughly speaking (precise definitions will be given in the next section), a Beauville surface is a complex surface S defined by taking a pair of complex curves, i.e., Riemann surfaces, C1 and C2 and letting a finite group G act freely on their product to define S as a quotient (C1 × C2 )/G. These surfaces have a wide variety of attractive geometric properties: they are surfaces of general type; their automorphism groups [45] and fundamental groups [6, 18] are relatively easy to compute (being closely related to G); these surfaces are rigid surfaces in the sense of admitting no nontrivial deformations [8] and thus correspond to isolated points in the moduli space of surfaces of general type [19, 32]. Much of this good behaviour stems from the fact that the surface (C1 × C2 )/G is uniquely determined by a particular pair of generating sets of G known as a ‘Beauville structure’. This converts the study of Beauville surfaces to the study of groups with Beauville structures, i.e., Beauville groups. Beauville surfaces were first defined by Catanese in [18] as a generalisation of an earlier example of Beauville [12, Exercise X.13(4)] (native English speakers may find the English translation [13] somewhat easier to read and get hold of) in which C = C  and the curves are both the Fermat curve defined by the equation X 5 + Y 5 + Z 5 = 0 being acted on by the 5-group C5 × C5 (we write Cn for the cyclic group of order n. This choice of group may seem somewhat odd at first, but the reason will become clear later). Bauer, Catanese and Grunewald went on to use these surfaces to construct examples of smooth regular surfaces with vanishing geometric genus [9]. Early motivation came from the consideration of the ‘Friedman-Morgan speculation’ — a technical conjecture concerning when two algebraic surfaces are diffeomorphic which Beauville surfaces provide counterexamples to. More recently, they have been used to construct interesting orbits of the absolute Galois group Gal(Q/Q) [35, 38, 39] (connections with Gothendeick’s theory of dessins d’enfant make it possible for this group to act on the set of all

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Beauville surfaces). Furthermore, Beauville’s original example has also been used by Galkin and Shinder in [31] to construct examples of exceptional collections of line bundles. Whilst these constructions work for finite groups in general there is particular interest in studying the special case of p-groups. First note that in some sense ‘most finite groups are p-groups’: there are 49 910 529 484 groups of order at most 2 000. Of these groups 49 487 365 422 have order precisely 1 024 — that is more than 99 · 1% of the total! When we add to this collection all the other p-groups of order at most 2 000 we have essentially all of them. By comparison the total number of finite simple groups of order at most 2000 is merely six! Answering questions about Beauville groups in the case of p-groups thus goes a long way to answering these questions for finite groups in general. (For details of the extraordinary and impressive computational feats just mentioned and a general historical discussion of the problem of enumerating groups of small order, which has been worked on for almost a century and a half, see the work of Besche, Eick and O’Brien in [15, 16].) Moreover, as we shall see later, there are many reasons for believing that Beauville p-groups with various properties are much harder to construct compared to other cases. Like any survey article, the topics discussed here reflect the research interests of the author. In many ways this survey is the sequel to the author’s contribution to the last Groups St Andrews proceedings [22] though the reader will lose little if they have neither read nor have to hand a copy of [22]. Slightly older surveys discussing related geometric and topological matters are given by Bauer, Catanese and Pignatelli in [11, 10]. Other notable works in the area include [5, 46, 54, 57]. This survey is organised as follows. In Section 2 we will give precise definitions for the various Beauville constructions described in vague terms above. In Section 3 we will describe general constructions of Beauville p-groups before moving on in Section 4 to focus on the so-called ‘mixed case’ and in Section 5 we will focus on strongly real examples. Finally, in Section 6 we discuss open questions, problems and conjectures in the area.

2

Main Definitions

Definition 2.1 A surface S is a Beauville surface of unmixed type if • the surface S is isogenous to a higher product, that is, S ∼ = (C1 × C2 )/G where

C1 and C2 are algebraic curves of genus at least 2 and G is a finite group acting faithfully on C1 and C2 by holomorphic transformations in such a way that it acts freely on the product C1 × C2 , and

• each Ci /G is isomorphic to the projective line P1 (C) and the covering map Ci → Ci /G is ramified over three points. There also exists a concept of Beauville surfaces of mixed type but we shall postpone our discussion of these until Section 4. In the first of the above conditions the genus of the curves in question needs to be at least 2. It was later proved by Fuertes, Gonz´alez-Diez and Jaikin-Zapirain in [29] that in fact we can take

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the genus as being at least 6. The second of the above conditions implies that each Ci carries a regular dessin in the sense of Grothendieck’s theory of dessins d’enfants (childeren’s drawings) [40]. Furthermore, by Bely˘ı’s Theorem [14] this ensures that S is defined over an algebraic number field in the sense that when we view each Riemann surface as being the zeros of some polynomial we find that the coeffcients of that polynomial belong to some number field. Equivalently they admit an orientably regular hypermap [49], with G acting as the orientation-preserving automorphism group. A modern account of dessins d’enfants and proofs of Bely˘ı’s theorem may be found in the recent book of Girondo and Gonz´ alez-Diez [34]. An even more recent account of dessins d’enfants, which culminates in a final chapter on Beauville surfaces, is given by Jones and Wolfart in [50]. These surfaces can also be described instead in terms of uniformisation and the language of Fuchsian groups [37, 56]. What makes this class of surfaces so good to work with is the fact that all of the above definition can be ‘internalised’ into the group. It turns out that a group G can be used to define a Beauville surface if and only if it has a certain pair of generating sets known as a Beauville structure. Definition 2.2 Let G be a finite group. Let x, y ∈ G and let Σ(x, y) :=

|G|  

{(xi )g , (y i )g , ((xy)i )g }.

i=1 g∈G

An unmixed Beauville structure for the group G is a set of pairs of elements {{x1 , y1 }, {x2 , y2 }} ⊂ G × G with the property that x1 , y1 = x2 , y2 = G such that Σ(x1 , y1 ) ∩ Σ(x2 , y2 ) = {e}. If G has a Beauville structure we say that G is a Beauville group. Furthermore we say that the structure has type ((o(x1 ), o(y1 ), o(x1 y1 )), (o(x2 ), o(y2 ), o(x2 y2 ))). Historically, authors have defined the above structure in terms of so-called ‘spherical systems of generators of length 3’, meaning {x, y, z} ⊂ G with xyz = e, but we omit z = (xy)−1 from our notation in this survey. (The reader is warned that this terminology is a little misleading since the underlying geometry of Beauville surfaces is hyperbolic thanks to the below constraint on the orders of the elements.) Furthermore, many earlier papers on Beauville structures add the condition that for i = 1, 2 we have that 1 1 1 + + < 1, o(xi ) o(yi ) o(xi yi ) but this condition was subsequently found to be unnecessary following Bauer, Catanese and Grunewald’s investigation of the wall-paper groups in [7]. A triple of

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elements and their orders satisfying this condition are said to be hyperbolic. Geometrically, the type gives us considerable amounts of geometric information about the surface: the Riemann-Hurwitz formula   1 1 1 |G| g(Ci ) = 1 + 1− − − 2 o(xi ) o(yi ) o(xi yi ) tells us the genus of each of the curves used to define the surface S and by a theorem of Zeuthen-Segre this also gives us the Euler number of the surface S since e(S) = 4

(g(C1 ) − 1)(g(C2 ) − 1) |G|

which in turn gives us the holomorphic Euler-Poincar´e characteristic of S, namely 4χ(S) = e(S) (see [18, Theorem 3.4]). Already we see one reason for finding the p-groups case more challenging — in many places in the literature a commonly used trick to show that the condition that Σ(x1 , y1 ) ∩ Σ(x2 , y2 ) = {e} is satisfied is to find a Beauville structure such that o(x1 )o(y1 )o(x1 y1 ) is coprime to o(x2 )o(y2 )o(x2 y2 ) but this clearly cannot be done in a p-group since every non-trivial element has an order that is a power of p. There are several properties a p-group may have that make it intuitively more likely to be a Beauville group. Having a low exponent makes it easier for Σ(g, h) to be small, indeed it is not difficult to prove that if n > 2 then a Beauville p-group of order pn must have exponent at most pn−2 (although there do exist Beauville p-groups attaining this bound). Moreover, having a large abelian subgroup, and in particular a large center, makes it easier for elements to have large centralizers and thus belong to small conjugacy classes, again making Σ(g, h) small.

3

General Constructions

The earliest examples of Beauville p-groups were given in Catanese’s original paper [18] where he showed that the groups Cn × Cn are Beauville groups whenever n > 1 is coprime to 6. (Later in [7] Bauer, Catanese and Grunewald showed that these are in fact the only abelian Beauville groups. A proof adapted from theirs is given by Jones and Wolfart in [50, Theorem 11.1].) In particular if p > 3 is a prime, then we can take n to be a power of p giving infinitely many examples of (abelian) Beauvillle p-groups, though alas this also tells us that there are no abelian examples at all when p = 2 or 3. This explains Beauville’s original choice for his example — C5 × C5 is the smallest abelian group that is a Beauville group. Subsequently in [36] Gonz´alez-Diez, Jones and Torres-Teigell put this classification to great use deriving a number of interesting facts about the surfaces associated with these groups. For example they showed that the field of definition of the corresponding surfaces is always Q. They also showed that if q is a power of a prime p > 3, then the number of isomorphism classes of Beauville surfaces coming from the Beauville group Cq × Cq is asymptotically q 4 /72. The earliest examples of Beauville 2-groups and 3-groups (that are unmixed — we postpone a brief discussion of some slightly older mixed Beauville 2-groups until

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the next section) were given by Fuertes, Gonz´ alez-Diez and Jaikin-Zapirain in [29, Section 5] where isolated examples of Beauville groups of order 212 and 312 are constructed. The earliest systematic attempt to construct infinite families of Beauville pgroups was by Barker, Boston and the author in [1] where they considered groups of order pn for small values of n. In particular they found the smallest Beauville p-group for every p by proving the following [1, Corollary 9]. Lemma 3.1 The smallest non-abelian Beauville p-groups are as follows: (a) for p = 2 the group of order 27 defined by the presentation u, v | (ui v j )4 for i, j = 0, . . . , 3, (u2 v 2 )2 , [u, v]2 , (uvuv 3 )2 (b) for p = 3 the group of order 35 defined by presentation x, y, z, w, t | x3 , y 3 , z 3 , w3 , t3 , y x = yz, z x = zw, z y = zt where whenever any two of the generators, g and h, commute we omitted the relation [g, h] = e for clarity and (c) for p ≥ 5 the extraspecial group p1+2 defined by the presentation + x, y, z | xp , y p , z p , yx = xyz, [x, z], [y, z] . They went on to show that for every p ≥ 5 and n ≥ 4 there exists a non-Beauville group of order pn and there exists at least one Beauville group of order pn . They then went on to completely classify the Beauville groups of order at most p4 and go most of the way to classifying the Beauville groups of order p5 and p6 (more on this later). One interesting feature of this work was the observation that as p → ∞ the proportion of 2-generated groups of order p5 that are Beauville tends to 1, however as p → ∞ the proportion of 2-generated groups of order p6 that are Beauville does not tend to 1! In [4] the Barker, Boston, Peyerimhoff and Vdovina, for reasons that will become slightly clearer in the next section, considered quotients of the group x0 , x1 , . . . , x12 | x3i , xi xi+1 xi+4 for i = 0, . . . , 13 (the indices in the relations should be read modulo 13) and the index 3 subgroup generated by x0 , x1 and x2 . Doing this they were able to construct a finite number of Beauville 3-groups, conjecturing that an infinite family can be obtained this way. As an interesting application of these ideas in [52] Peyerimhoff and Vdovina were able to obtain expander graphs (infinite families of finite graphs that are sparse and yet simultaneously highly connected, which enjoy applications in computer science for example in the design of highly robust networks) from the Cayley graphs corresponding to generating sets for these groups. More recently in [27] Fern´ andez-Alcober and G¨ ul took a rather different approach. First they prove the following general result for constructing Beauville p-groups when p ≥ 5.

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Theorem 3.2 Let G be a 2-generator finite p-group of exponent pe and suppose that G satisfies one of the following conditions (a) For x, y ∈ G, we have xp

e−1

= yp

e−1

if and only if (xy −1 )p

e−1

= e.

(b) G is a potent p-group. e−1

Then G is a Beaville group if and only if p ≥ 5 and |Gp | ≥ p2 . If that is the case, then every lift of a Beauville structure of G/Φ(G) is a Beauville structure of G. Here a finite p-group is said to be ‘potent’ if either p > 2 and γp−1 (G) ≤ Gp or p = 2 and G ≤ G4 where γi (G) is inductively defined by the lower central series γ1 (G) = G and for i > 1 we have that γ(G)i = [γi−1 (G), G]. In later parts of [27] Fern´andez-Alcober and G¨ ul consider quotients of the famous Nottingham group to construct more infinite families of p-groups for p odd and in particular gave the first infinite family of Beauville 3-groups. In [17] Boston discussed Beauville p-groups in terms of the ‘O’Brien tree’ of p-groups. Given a 2-generated p-group with lower p-central series of subgroups (Pi (G))ni=1 (recall that this is defined inductive by P0 (G) = G and for i > 0, Pi+1 (G) = [G, Pi (G)]Pi (G)p ) we have a sequence of quotients G = G/Pn (G) → G/Pn−1 (G) → · · · → G/P1 (G) = Cp2 . For a fixed p we define a tree whose vertices are the isomorphism classes of 2generated p-groups (the most general construction drops the 2-generated restriction, but in this setting it seems appropriate to make this assumption). A p-group G of class i is then adjoined to any p-group H of class i + 1 such that H/Pi (H) ∼ =G (these are called the ‘children’ of G). Boston investigated the relationship between a p-group’s status as a Beauville group and the status of its children surprisingly finding little correlation between the two and even finding p-groups with infinitely many children none of which are Beauville. There does, however appear to be some connection with a quantity associated with the O’Brien tree known as the ‘nuclear rank’ of a p-group. See [51] for the technical details. The following general lemma of Fuertes and Jones [30, Lemma 4.2] (originally proved with special linear groups in mind) has been applied in various parts of the literature to Beauville p-groups. Lemma 3.3 If x, y, u, v ∈ G have images x ¯, y¯, u ¯, v¯ ∈ G/N yielding a Beauville structure in G/N , then if x ∩ N = y ∩ N = z ∩ N = {e}, then x, y, u, v give a Beauville structure in G. In particular in [17, Corollary 3.3] deduces the following. Corollary 3.4 If p ≥ 5 and Γ is the triangle group x, y, z | xp = y p = z p = xyz = e , then its p-central quotients Γ/Pi (Γ) are all Beauville groups.

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On the basis of this Boston conjectured that if the group Γ is either the free product x, y | xp , y p or the free group of rank 2, then Γ/Pi (Γ) are all Beauville. Both cases of this conjecture were later proved in the affirmative in [42] by G¨ ul who also noted that when comparing the infinite family of Beauville 3-groups this gives with the family discussed above that were constructed in [27] the only group lying in both families was the smallest one, namely the Beauville group of order 35 as given in Lemma 3.1. More recently in [26] Fern´ andez-Alcober, Gavioli, G¨ ul and Scoppola proved a related result similar to Lemma 3.3 along the following lines. If G is a Beauville p-group that behaves well with respect to taking powers then every Beauville structure of G/Φ(G) is a Beauville structure of G. They call a Beauville p-group ‘wild’ if it fails to have this property and classify the matabelian Beauville p-groups and p-groups with a maximal subgroup of class at most two. In doing so they exhibit infinitely many Beauville p-groups of wild type. In [55] Stix and Vdovina took a more global view of all Beauville p-groups by deploying the theory of pro-p groups (leading to the introduction of a new notion, that of a ‘topological Beauville structure’) to prove the following. Theorem 3.5 Every finite p-group with an unmixed Beauville structure sits in an infinite pro-system of p-groups with compatible unmixed Beauville structure such that the type of the first half remains constant throughout the pro-system. In the same paper they were able to obtain a new infinite family of non-abelian Beauville p-groups by proving the following. Theorem 3.6 Let m, n ∈ N and λ ∈ (Z/pm Z)× with λp semidirect product Z/pm Z :λ Z/pn Z

n

≡ 1 mod pm . The

with action Z/pn Z → Aut(Z/pm Z) sending 1 → λ admits an unmixed Beauville structure if and only if p ≥ 5 and m = n. One of the most recent constructions of Beauville p-groups comes from the following general criterion given by Jones and Wolfart in [50, Chapter 11]. Theorem 3.7 Let G be a finite group of exponent n = pe > 1 for some prime p ≥ 5, such that the abelianisation G/G of G is isomorphic to Cp × Cp . Then G is a Beauville group. Corollary 3.8 Let G be a 2-generated finite group of exponent p for some prime p ≥ 5. Then G is a Beauville group. To give concrete examples of groups satisfying the hypotheses of these results they leave to the reader the exercise of showing that if W is the wreath product Cn  Cn , then the quotient of this group by the ‘diagonal subgroup’ (i.e., the center) is a group that satisfies the hypotheses of Theorem 3.7 though they also remark that “Since p-groups tend to have many quotients, these results show that there is no shortage of groups satisfying the hypotheses of [these results].”

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Another recent construction is given by Gaviloli, G¨ ul and Scoppola in [33] which we describe as follows. A non-cyclic finite p-group G is said to be ‘thin’ if every normal subgroup of G lies between two consecutive terms of the lower central series and |γi (G) : γi+1 (G)| ≤ p2 for all i ≥ 1. A classification of which of these are Beauville is given in [33], the precise statement splitting into a number of different cases. Here we have discussed general constructions of Beauville p-groups that (at the time of writing) are not known to have any particular additional properties from the viewpoint of being Beauville groups. In the next two section we discuss constructions of Beauville p-groups that are known to have additional properties, namely the properties of being ‘mixed’ and the property of being ‘strongly real’.

4

The Mixed Case

When we defined Beauville surfaces and groups we considered the action of a group G on the product of two curves C1 × C2 . In an unmixed structure this action comes solely from the action of G on each curve individually, however there is nothing to stop us considering an action on the product that interchanges the two curves and it is precisely this situation that we discuss in this section. Recall from Definition 2.2 that given x, y ∈ G we write Σ(x, y) :=

|G|  

{(xi )g , (y i )g , ((xy)i )g }.

i=1 g∈G

Definition 4.1 Let G be a finite group. A mixed Beauville structure for G is a quadruple (G0 , g, h, k) where G0 is an index 2 subgroup and g, h, k ∈ G are such that • g, h = G0 ; • k ∈ G0 ; • for every γ ∈ G0 we have that (kγ)2 ∈ Σ(g, h) and • Σ(g, h) ∩ Σ(g k , hk ) = {e} A Beauville surface defined by a mixed Beauville structure is called a mixed Beauville surface and a group possessing a mixed Beauville structure is called a mixed Beauville group. In terms of the curves defining the surface, the group G0 is the stabiliser of the curves with the elements of G \ G0 interchanging the two terms of C1 × C2 . Moreover it is only possible for a Beauville surface (C1 × C2 )/G to come from a mixed Beauville structure if C1 ∼ = C2 . The above conditions also ensure that {{g, h}, {g k , hk }} ⊂ G0 × G0 is a Beauville structure for G0 . In general, mixed Beauville structures are much harder to construct than their unmixed counterparts and, as noted in the introduction, it is even harder still in the case of p-groups: since a mixed Beauville group necessarily has an index 2 subgroup we must have p = 2. In particular, this fact combined with Catanese’s

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classification of the abelian Beauville groups that we discussed at the beginning of Section 3 we can immediately see that there are no abelian mixed Beauville groups. The following lemma of Fuertes and Gonz´ alez-Diez imposes a strong condition on a group with a mixed Beauville structure [28, Lemma 5]. Lemma 4.2 Let (C1 × C2 )/G be a mixed Beauville surface and let G0 be the subgroup of G consisting of the elements which do not interchange the two curves. Then the order of any element in G \ G0 is divisible by 4. Some of the earliest examples of Beauville 2-groups were constructed in an effort to find examples of mixed Beauville groups. In [9] Bauer, Catanese and Grunewald reported computer calculations they had done to find small examples verifying that there were none of order strictly less than 28 and only two such groups of order 28 which have the same index 2 subgroup. Explicitly the index 2 subgroup is the one mentioned in part (a) of Lemma 3.1. The groups themselves are given by the presentations x1 , . . . , x8 | x21 = x4 x5 x6 , x22 = x4 x5 , x23 = x4 , x24 = x25 = x26 = x27 = x28 = e xx2 1 = x2 x4 , xx3 1 = x3 x5 , xx3 2 = x3 x6 , xx4 1 = x4 x7 , xx4 2 = x4 x8 , xx5 1 = x5 x7 x8 , xx5 2 = x5 x8 , xx5 3 = x5 x7 , xx6 1 = x6 x8 , xx6 2 = x6 x7 , xx6 3 = x6 x8 and x1 , . . . , x8 | x21 = x4 x5 x6 x7 , x22 = x4 x5 , x23 = x4 , x24 = x25 = x26 = x27 = x28 = e xx2 1 = x2 x4 , xx3 1 = x3 x5 , xx3 2 = x3 x6 , xx4 1 = x4 x7 , xx4 2 = x4 x8 , xx5 1 = x5 x7 x8 , xx5 2 = x5 x8 , xx5 3 = x5 x7 , xx6 1 = x6 x8 , xx6 2 = x6 x7 , xx6 3 = x6 x8 . the index 2 subgroup being the group defined in Lemma 3.1. Six more examples of orders 214 , 216 , 219 , 224 , 227 were constructed by Barker, Boston, Peyerimhoff and Vdovina in [2] where they conjectured that their method could be adapted to give an infinite family of mixed Beauville 2-groups. They later verified this to be correct in [3] by constructing an infinite family of Beauville 2-groups as quotients of the group x0 , x1 , . . . , x6 | xi xi+1 xi+3 for i = 0, . . . , 6 (the indices should be read modulo 7) and the index 2 subgroup generated by x0 and x1 only. The relations in the above presentation should immediately make the reader think of the famous Fano plane and generalising in this direction is indeed a subsequent development of the subject it being just one example of a ‘group with special presentation’ defined this way. This gave the first infinite family of

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mixed Beauville 2-groups that were constructed by Barker, Boston, Peyerimhoff and Vdovina in [3]. As far as the author is aware, the most recent general discussion of examples of mixed Beauville groups is given by the author and Pierro in [25] (though none of the Beauville groups constructed in [25] are even soluble, let alone are p-groups).

5

The Strongly Real Case

Given any complex surface S it is natural to consider the complex conjugate surface S. In particular it is natural to ask if the surfaces are biholomorphic. Definition 5.1 Let S be a complex surface. We say that S is strongly real if there exists a biholomorphism σ : S → S such that σ ◦ σ is the identity map. As noted earlier this geometric condition can be translated into algebraic terms. Definition 5.2 Let G be a Beauville group and let X = {{x1 , y1 }, {x2 , y2 }} be a Beauville structure for G. We say that G and X are strongly real if there exists an automorphism φ ∈ Aut(G) and elements gi ∈ G for i = 1, 2 such that and gi φ(yi )gi−1 = yi−1 . gi φ(xi )gi−1 = x−1 i It is often, but not always, convenient to take g1 = g2 = e. Again what makes the case of p-groups particularly interesting is how much harder it is to construct strongly real examples in this case compared to groups in general, especially when p is odd. In [44] Helleloid and Martin prove that automorphism group of a finite p-group is almost always a p-group. In particular, if p is odd, then typically no automorphism like the φ in of Definition 5.2 exists since such an automorphism must necessarily have even order. Nonetheless many examples have been found. We first note that every abelian Beauville group is strongly real since the function x → −x is always an automorphism of an abelian group and in particular the abelian Beauville p-groups discussed at the beginning of Section 3 are strongly real. As far as the author is aware the earliest examples of non-abelian strongly real Beauville p-groups to be discovered were an isolated pair of examples of 2-groups constructed by the author in [20, Section 7] namely the groups u, v | (ui v j )4 , i, j = 0, 1, 2, 3 which has order 214 and u, v | u8 , v 8 , [u2 , v 2 ], (ui v j )4 , i, j = 1, 2, 3 which has order 213 . We take this opportunity to correct an error made by the author in the original proof of [20, Lemma 3] where this was first proved. In the original proof it is stated that we can take x1 := u, y1 := v, x2 := uvu

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and y2 := vuv as our generators however these elements clearly cannot provide a Beauville structure because x2 y2 ∈ Σ(x1 , y1 ) since x2 y2 = (x1 y1 )3 . If, however, we instead take y2 := uvuvu then the function mapping u ↔ u−1 and v ↔ v − 1 still inverts this new y2 . It is easy to see that x2 , y2 = G since vu = x−1 2 y2 ; v = x2 (vu)−1 and u = v −1 (vu) hence u, v ∈ x2 , y2 . The other conditions of being a Beauville structure are also easily checked by computer. Recently in [41] G¨ ul constructed the first known infinite family of non-abelian strongly real Beauville p-groups and in particular discovered the first examples in which p is odd. More specifically, the main result of [41] is the following. Theorem 5.3 Let F = x, y | xp , y p be the free product of two cyclic groups of order p for an odd prime p and let i = k(p − 1) + 1 for k ≥ 1. Then the quotient F/γi+1 (F ) is a strongly real Beauville group. Subsequently in [43] G¨ ul constructed further examples by considering quotients of certain triangle groups. More specifically G¨ ul prove that there are non-abelian strongly real Beauville p-groups of order pn for every n ≥ 3, 5 or 7 for the primes p ≥ 5, p = 3 and p = 7 respectively. At around the same time the author constructed another infinite family of nonabelian strongly real Beauville p-groups for p odd in [23, 24] by proving the following. Theorem 5.4 Let p be an odd prime and let q and r be powers of p. If q and r are sufficiently large, then groups Cq  Cr /Z(Cq  Cr ) are strongly real Beauville groups. Unlike the groups given by Theorem 5.3 this theorem gives multiple non-isomorphic examples for infinitely many orders. For example when (q, r) = (328 , 33 ) or (q, r) = (33 , 35 ) we obtain groups of order 3731 which cannot be isomorphic since they have centers of different orders. As far as the author is aware the best references for what is known about strongly Beauville groups more generally are the surveys given by the author in [20, 21].

6

Open Questions, Problems and Conjectures

6.1

Field of Definition of Beauville Surfaces

In Section 3 we mentioned the results of [36] due to Gonz´ alez-Diez, Jones and Torres-Teigell on surfaces defined by abelian Beauville p-groups with p ≥ 5. Determining the field of definition of a Beauville surface is in general a difficult problem. The next most easy cases after abelian p-groups seem to be the smallest non-ableian Beauville p-groups as classified by Barker, Boston and the author in [1]. Question 6.1 What are the fields of definition of the Beauville surfaces defined by the smallest Beauville p-groups? A related question is the following. Question 6.2 Does being defined by a Beauville p-group have geometric consequences for the underlying surface and if so what are they?

282 6.2

Fairbairn: Beauville p-groups: a survey Beauville Dimension

In unpublished correspondence with the author Gareth A. Jones has recently introduced the intriguing notion ‘Beauville dimension’ that we define as follows. Definition 6.3 Let G be a finite group. A group G is said to have Beauville dimension n if there exist n pairs of elements x1 , . . . , xn , y1 , . . . , yn ∈ G such that xi , yi = G for every i = 1, . . . , n; n

Σ(xi , yi ) = {e}

i=1

and no set of n − 1 pairs of generators can be found with this property. If no such n exists then we say that G has infinite Beauville dimension. By way of examples from p-groups, clearly any Beauville group has Beauville dimension 2 whilst dihedral 2-groups all have infinite Beauville dimension since for any generating pair g and h the set Σ(g, h) will necessarily contain the centeral involution. (Another, albeit non-nilpotent, example is the alternating group A5 : in this case Σ(g, h) necessarily contains elements from the only class of cyclic subgroup of order 5.) What makes this concept interesting is as follows. Having Beauville dimension n means that the group G can act on a product of n Riemann surfaces with many of the nice geometric properties enjoyed by Beauvllie surfaces that we discussed in Section 1 also being enjoyed by these higher dimensional manifolds and varieties (thanks to Serre’s GAGA principal they will be both varieties and manifolds). In short, this enables a higher dimensional theory of Beauville-like constructions. As far as the author is aware the only known examples of groups with finite Beauville dimension greater than 2 are as follows. In personal correspondence Jones observed that the 3-groups C3a × C3a all have Beaville dimension 4, an easy exercise for the reader. Based on this, recent work of the author’s PhD student Ludo Carta, has produced infinitely many examples with Beauville dimension 3 and 4. These largely come from the unusual behavior for certain 3-groups. All of this immediately poses the following question. Question 6.4

(a) Do there exist groups of Beauville dimension n > 4?

(b) Is the Beauville dimension of 2-generated finite groups bounded or can it get arbitrarily large? Whilst this is not a question that specifically focuses on p-groups per se, it does seem that p-groups, in particular 3-groups, are a fertile breeding ground for examples that will address the above question. Further reason for believing that this is really a question concerning p-groups is the fact that the only finite quasisimple groups that do not have Beauville dimension 2 are the alternating group A5 and its covering group SL2 (5) both of which have infinite Beauville dimension. Characteristically simple groups similarly seem to be (and are conjectured to be — see [47, 48]) Beauville whenever they are 2-generated whilst something similar is

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true for almost simple groups. It follows that non-nilpotent examples will be very likely to be at best soluble. 6.3

What proportion of 2-generated p-groups are Beauville groups?

As previously mentioned in Section 3 the question of what proportion of 2-generated p-groups are Beauville groups as raised by Barker, Boston and the author in [1] is particularly interesting since it seems that ‘most’ groups of order p5 are Beauville whereas we cannot say the same things about groups of order p6 . It is natural to ask which of these two situations are typical of groups of order pn for general n. Question 6.5 For which n does the proportion of 2-generated groups of order pn that are Beauville tend to 1? When it does not tend to 1 does it tend to 0? One need only look at the number of groups of order at most p5 to realise this is the point where the number of groups of order pn suddenly starts depending on p and it just happens that n = 5 is sufficiently tame for the proportion to tend to 1, however as n increases the formula for the number of groups of order pn becomes even more intertwined with the value of p. In the opinion of the author it is unlikely that the proportion of 2-generated groups of order pn that are Beauville groups will tend to 1 as p tends to infinity for any n > 5. 6.4

Are Beauville groups typically strongly real?

As mentioned in Section 1 one motivation for focusing on the p-groups case is that in some sense ‘most’ finite groups are p-groups. It follows that the general question of what proportion of 2-generated p-groups are strongly real Beauville groups naturally translates to one of focusing on p-groups. The following is posed by the Author in [23, Section 3]. Question 6.6 (a) How does the proportion of Beauville groups of order pn that are strongly real vary as n increases? (b) How does the proportion of Beauville groups of order pn that are strongly real vary as p increases? In the opinion of the author the aforementioned work of Helleloid and Martin in [44] suggests that most Beauville 2-groups are strongly real and that for p odd very few Beauville p-groups are strongly real. A related question is the following. One of the first questions to be asked about Beauville groups was, given a Beauville group, can we enumerate its Beauville structures? This raises the related question regarding strongly real Beauville groups. Question 6.7 Given a strongly real Beauville p-group, what proportion of its Beauville structures are strongly real?

284 6.5

Fairbairn: Beauville p-groups: a survey Groups of order pn for small n

In [1, 27] the Beauville groups of order at most p6 are classified, however the pgroups of order at most pn for larger values of n are known. This poses the following natural question. Problem 6.8 Classify the Beauville groups of order pn for n > 6. Since the full classification of groups of order for pn is only known as far as p9 (with the exception of groups of order 210 that have also been classified — see [15, 16]) answering this question even for modest values of n is a computationally intensive task that is unlikely to be completed any time soon. 6.6

Beauville Spectra

The following definition was first made by Fuertes, Gonz´alez-Diez and JaikinZapirain in [29, Definition 11]. Definition 6.9 Let G be a finite group. The Beauville genus spectrum of G, denoted Spec(G), is the set of pairs of integers (g1 , g2 ) such that g1 ≤ g2 and there are curves C1 and C2 of genera g1 and g2 with the action of G on C1 × C2 such that (C1 × C2 )/G is a Beauville surface. They went on in [29] to determine the Beauville genus spectra for the symmetric group S5 the linear group L2 (7) and abelian Beauville groups as well as showing that Spec(S6 ) = ∅ (though clearly this last result has been generalised by any theorem proving that another group is a Beauville group). These calculations were later pushed further to other small almost-simple groups by Pierro in his PhD thesis [53], the largest group he considered being the Matheiu group M11 (whose order is just 7 092) there being 87 such pairs for this group. As the orders of the groups grow the size of the Beauville genus spectrum grows too making it difficult to push these calculations for for almost-simple groups much further. If a group is a p-group, however, the story is very different. Since there is a much narrower range of element orders (being small powers of p) and it is the orders of the elements in a Beauville structure that determine the genera of the corresponding curves, there is a much narrower range of possibilities for the Beauville spectrum making the task of finding them much more managable, especially given that Beauville p-groups tend to have low exponent. For example the following is immediate. Lemma 6.10 The Beauville genus spectrum of a 2-generated p-group of exponent p is

  (p − 1)(p − 2) (p − 1)(p − 2) , . 2 2 By way of example, for p ≥ 5 the groups Cp ×Cp and the extraspecial groups p1+2 + are all groups that satisfy the hypotheses of the above lemma. This immediately raises the following interesting problem.

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Problem 6.11 Determine the Beauville genus spectrum of Beauville p-groups. We also introduce the following. Definition 6.12 The strongly real Beauville genus spectrum of G, that we shall denote SRSpec(G) is the set of pairs of integers (g1 , g2 ) such that g1 ≤ g2 and there are curves C1 and C2 of genera g1 and g2 with the action of G on C1 × C2 such that (C1 × C2 )/G is a real Beauville surface. Since elements of larger order tend to have the property that no automorphism will map them to their inverses it seems likely to the author that SRSpec(G) will in general be much smaller than Spec(G) for most groups. In particular, if determining Spec(G) for a given group G is difficult owing to its size, the problem performing the same task for SRSP ec(G) may be much more tractable. Problem 6.13 Determine the strongly real Beauville genus spectrum of Beauville p-groups. As noted earlier every Beauville structure of an abelian Beauville group is necessarily strongly real so for these groups we must have Spec(G) = SRSpec(G). For non-abelian Beauville p-groups it is likely that |SRSpec(G)| < |Spec(G)|. This motivates the following interesting question. Question 6.14 For a Beauville p-group G how does the size of SRSpec(G) compare to Spec(G)? A little more specifically, how does |SRSpec(G)|/|Spec(G)| behave as |G| → ∞? References [1] N. W. Barker, N. Boston and B. T. Fairbairn, A note on Beauville p-groups, Exp. Math. 21 (2012), no. 3, 298–306. [2] N. W. Barker, N. Boston, N. Peyerimhoff and A. Vdovina, New examples of Beauville surfaces, Monatsh. Math. 166 (2012), no. 3–4, 319–327. [3] N. W. Barker, N. Boston, N. Peyerimhoff and A. Vdovina, An infinite family of 2groups with mixed Beauville structures, Int. Math. Res. Not. IMRN 2015, no. 11, 3598–3618. [4] N. W. Barker, N. Boston, N. Peyerimhoff and A. Vdovina, Regular algebraic surfaces, ramification structures and projective planes, in Beauville Surfaces and Groups (eds. I. Bause, S. Garion and A. Vdovina), 15–33, Springer Proc. Math. Stat., 123, Springer, 2015. [5] I. Bauer, Product-Quotient Suraces: Result and Problems, preprint 2012. arxiv:1204.3409 [6] I. Bauer, F. Catanese and D. Frapporti, The fundamental group and torsion group of Beauville surfaces, in Beauville Surfaces and Groups (eds. I. Bauer S. Garion and A. Vdovina), 1–14 (2015). [7] I. Bauer, F. Catanese and F. Grunewald, Beauville surfaces without real structures, in Geometric methods in algebra and number theory, 1–42, Progr. Math., 235, Birkh¨auser Boston, Boston, MA, 2005. [8] I. Bauer, F. Catanese and F. Grunewald, Chebycheff and Belyi polynomials, dessins d’enfants, Beauville surfaces and group theory, Mediterr. J. Math. 3 (2006), 121–146.

286

Fairbairn: Beauville p-groups: a survey

[9] I. Bauer, F. Catanese and F. Grunewald, The classification of surfaces with pg = q = 0 isogenous to a product of curves, Pre Appl. Math. Q. 4 (2008), no. 2, Special Issue: In honor of Fedor Bogomolov. Part 1, 547–586. [10] I. C. Bauer, F. Catanese and R. Pignatelli, Complex surfaces of general type: some recent progress, in Global aspects of complex geometry, 1–58, Springer, Berlin, 2006. [11] I. Bauer, F. Catanese and R. Pignatelli, Surfaces of general type with geometric genus zero: a survey, in Complex and differential geometry, 1–48, Springer Proc. Math., 8, Springer-Verlag, Heidelberg, 2011. [12] A. Beauville, Surfaces alg´ebriques complexes, Ast´erisque 54, 1978. [13] A. Beauville, Complex Algebraic Surfaces, London Math. Soc. Student Texts 34, Cambridge University Press, Cambridge, 1996. [14] G. V. Bely˘ı, On Galois extensions of a maximal cyclotomic field, Math. USSR Izv. 14 (1980), 247–256. [15] H. U. Besche, B. Eick and E. A. O’Brien, The groups of order at most 2 000, Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 1–4. [16] H. U. Besche, B. Eick and E. A. O’Brien, A millennium project: constructing small groups, Internat. J. Algebra Comput. 12 (2002), no. 5, 623–644. [17] N. Boston, A survey of Beauville p-groups, in Beauville Surfaces and Groups (eds. I. Bauer, S. Garion and A. Vdovina), 35–40, Springer Proc. Math. Stat., 123, Springer, Cham, 2015. [18] F. Catanese, Fibered surfaces, varieties isogenous to a product and related moduli spaces, Amer. J. Math. 122 (2000), no. 1, 1–44. [19] F. Catanese, Moduli spaces of surfaces and real structures, Ann. of Math. (2) 158 (2003), no. 2, 577–592. [20] B. T. Fairbairn, Strongly real Beauville groups, in Beauville Surfaces and Group (eds. I. Bauer, S. Garion and A. Vdovina), 41–61, Springer Proc. Math. Stat., 123, Springer, Cham, 2015. [21] B. T. Fairbairn, More on strongly real Beauville groups, in Symmetries in Graphs, ˇ an Maps and Polytopes, (eds. J. Sir´ ˇ and R. Jajcay), 129–146, Springer Proc. Math. Stat., 159, Springer, 2016. [22] B. T. Fairbairn, Recent work on Beauville surfaces, structures and groups, in Groups St Andrews 2013 (eds. C. M. Campbell, M. R. Quick, E. F. Robertson and C. M. RoneyDougal), 225–241, London Math. Soc. Lecture Note Ser. 422, Cambridge University Press, 2015. [23] B. T. Fairbairn, A new infinite family of non-abelian strongly real Beauville p-groups for every odd prime p, Bull. Lond. Math. Soc. 49 (2017), no. 5, 749–654. [24] B. T. Fairbairn, A Corrigendum to “Fairbairn, A new infinite family of non-abelian strongly real Beauville p-groups for every odd prime p”, in preparation. [25] B. T. Fairbairn and E. Pierro, New examples of mixed Beauville groups, J. Group Theory 18 (2015), no. 5, 761-792. [26] G. A. Fern´andez-Alcober, N. Gavioli, S ¸ . G¨ ul and C. M. Scoppola, Beauville p-groups of wild type and groups of maximal class, preprint 2017. arXiv1701.07361. [27] G. A. Fern´ andez-Alcober and S¸. G¨ ul, Beauville structures in finite p-groups, preprint 2016. arXiv:1507.02942. [28] Y. Fuertes and G. Gonz´alez-Diez, On Beauville structures on the groups Sn and An , Math. Z. 264 (2010), no. 4, 959–968. [29] Y. Fuertes, G. Gonz´alez-Diez and A. Jaikin-Zapirain, On Beauville surfaces, Groups Geom. Dyn. 5 (2011), no. 1, 107–119. [30] Y. Fuertes and G. Jones, Beauville surfaces and finite groups, J. Algebra 340 (2011), 13–27. [31] S. Galkin and E. Shinder, Exceptional collections of line bundles on the Beauville

Fairbairn: Beauville p-groups: a survey

287

surface, Adv. Math. 244 (2013), 1033–1050. [32] S. Garion and M. Penegini, New Beauville surfaces, moduli spaces and finite groups, Comm. Algebra 42 (2014), no. 5, 2126–2155. [33] N. Gavioli, S¸. G¨ ul and C. Scoppola, Metabelian thin Beauville p-groups, preprint 2017. arXiv:1701.06906 [34] E. Girondo and G. Gonz´alez-Diez, Introduction to Compact Riemann Surfaces and Dessins d’Enfants, London Math. Soc. Student Texts 79, Cambridge University Press, Cambridge 2011. [35] G. Gonz´alez-Diez and A. Jaikin-Zapirain, The absolute Galois group acts faithfully on regular dessins and on Beauville surfaces, Proc. London Math. Soc. 111 (2015), no. 4, 775–796. [36] G. Gonz´alez-Diez, G. A. Jones and D. Torres-Teigell, Beauville surfaces with abelian Beauville group, Math. Scand. 114 (2014), no. 2, 191-204. [37] G. Gonz´alez-Diez and D. Torres-Teigell, An introduction to Beauville surfaces via uniformization, in Quasiconformal mappings, Riemann surfaces, and Teichm¨ uller spaces, 123–151, Contemp. Math., 575, Amer. Math. Soc., Providence, RI, 2012. [38] G. Gonz´alez-Diez and D. Torres-Teigell, Non-homeomorphic Galois conjugate Beauville structures on P SL(2, p), Adv. Math. 229 (2012), 3096–3122. [39] G. Gonz´ alez-Diez, G. A. Jones and D. Torres-Teigell, Arbitrarily large Galois orbits of non-homeomorphic surfaces, arXiv:1110.4930. [40] A. Grothendieck, Esquisse d’un Programme, in Geometric Galois Actions 1. in Around Grothendieck’s Esquisse d’un Programme (eds. P. Lochak and L. Schneps), 5–84, London Math. Soc. Lecture Note Ser. 242, Cambridge University Press, 1997. [41] S¸. G¨ ul, A note on strongly real Beauville p-groups, Monatsh. Math. (2017). [42] S¸. G¨ ul, Beauville structure in p-central quotients, J. Group Theory, to appear. [43] S¸. G¨ ul, An infinite family of strongly real Beauville p-groups, preprint 2016. arXiv:1610.06080. [44] G. T. Helleloid and U. Martin, The automorphism group of a finite p-group is almost always a p-group, J. Algebra 312 (2007), 284–329. [45] G. A. Jones, Automorphism groups of Beauville surfaces, J. Group Theory. 16 (2013), no. 3, 353–381. [46] G. A. Jones, Beauville surfaces and groups: a survey, in Rigidity and Symmetry (eds. R. Connelly, A. Weiss and W. Whitely), 205–225, Fields Inst. Commun., 70, Springer, New York, 2014. [47] G. A. Jones, Characteristically simple Beauville groups I: Cartesian powers of alternating groups, in Geometry, groups and dynamics, 289–306, Contemp. Math., 639, Amer. Math. Soc., Providence, RI, 2015. [48] G. A. Jones, Characteristically Simple Beauville Groups, II: Low Rank and Sporadic Groups, in Beauville Surfaces and Groups (eds. I. Bauer, S. Garion and A. Vdovina), 97–120, Springer Proc. Math. Stat., 123, Springer 2015. [49] G. A. Jones and D. Singerman, Belyi functions, hypermaps and Galois groups, Bull. Lond. Math. Soc. 28 (1996), 561–590. [50] G. A. Jones and J. Wolfart Dessins d’Enfants on Riemann Surfaces, Springer Monographs in Mathematics (2016). [51] E. A. O’Brien, The p-group generation algorithm, J. Symbolic Comput. 9 (1990), no. 5– 6, 677–698. [52] N. Peyerimhoff and A. Vdovina, Cayley graph expanders and groups of finite width, J. Pure Appl. Algebra 215 (2011), no. 11, 2780–2788. [53] E. Pierro, Some calculations on the action of groups on surfaces, PhD thesis, Birkbeck, University of London (2015). ˇ an [54] J. Sir´ ˇ, How symmetric can maps on surfaces be?, in Surveys in Combinatorics 2013

288

Fairbairn: Beauville p-groups: a survey

(eds. Simon R. Blackburn, Stefanie Gerke and Mark Wildon), 161–238, London Math. Soc. Lecture Note Ser., 409, Cambridge Univ. Press, Cambridge, 2013. [55] J. Stix and A. Vdovina, Series of p-groups with Beauville structure, Monatsh. Math. 181 (2016), no. 1, 177–186. [56] D. Torres-Teigell, Triangle groups, dessins d’enfants and Beauville surfaces, PhD thesis, Universidad Autonoma de Madrid, 2012. [57] J. Wolfart, ABC for polynomials, dessins d’enfants and uniformization — a survey, in Elementare und analytische Zahlentheorie, 313–345, Schr. Wiss. Ges. Johann Wolfgang Goethe Univ. Frankfurt am Main, 20, Franz Steiner Verlag Stuttgart, Stuttgart, 2006.

STRUCTURAL CRITERIA IN FACTORISED GROUPS VIA CONJUGACY CLASS SIZES ´ FELIPE, ANA MART´INEZ-PASTOR and V´ICTOR MANUEL MAR´IA JOSE ORTIZ-SOTOMAYOR∗ ∗

Instituto Universitario de Matem´ atica Pura y Aplicada (IUMPA-UPV), Universitat Polit`ecnica de Val`encia, Camino de Vera s/n, 46022, Valencia, Spain. Emails: [email protected], [email protected], [email protected]

Abstract We report on current activity regarding structural properties of finite factorised groups provided that the conjugacy class sizes of some elements in the factors have certain arithmetical conditions.

1

Introduction

All groups considered in this paper are finite. In recent decades, the analysis of groups which can be factorised as the product of two subgroups has been developed extensively, especially when those subgroups are linked by certain properties of permutability (see [5]). A main question in this topic is what can be said about the structure of the whole group when some information on the factors is known. On the other hand, several authors have carried out in-depth investigations with the purpose of understanding how the set of the conjugacy class sizes of a group influences its structure. An exhaustive report on this matter is due to Camina and Camina ([8]). A new research line arises when both current perspectives are joined together. More concretely, this paper is a survey article containing an upto-date account of recent achievements regarding factorised groups provided that the class sizes of elements in the factors verify certain arithmetical conditions. We shall show that previous results in [3, 7, 9, 10, 18] appear as corollaries when trivial factorisations are considered. Some earlier development in this setting can be found in [4, 16]. In the framework of factorised groups, it is well-known that the product of two normal supersoluble groups is not supersoluble in general. However direct and central products of supersoluble groups are so. Consequently, it seems reasonable to look into this problem under permutability assumptions stronger than normality (see [5]). One of the most relevant is the mutual permutability introduced in [2] by Asaad and Shaalan. It is said that two subgroups A and B of a group G are mutually permutable if A permutes with every subgroup of B and B permutes with every subgroup of A. Asaad and Shaalan showed that if a group G = AB is the product of the mutually permutable subgroups A and B and they both are supersoluble, then G is supersoluble provided that either the derived subgroup G of G is nilpotent or either A or B is nilpotent. In relation to this research, we show some results which supply supersolubility criteria for products of two (mutually

290 Felipe et al.: Structural criteria in factorised groups via conjugacy class sizes permutable) groups via conjugacy class sizes (see Section 2). In the sequel, when dealing with a factorised group G = AB, we only impose the arithmetical conditions on the class sizes in G of the elements in the factors A and B. Notice that, in such case, there is no relation in general between the size of the conjugacy class of an element in a factor and the size of its corresponding one in the whole group (see Examples 2.15, 3.10). Furthermore, we lose information about class sizes of elements of the form g = ab with a ∈ A and b ∈ B. There are easy examples which show that the class size of ab in G might not divide the product of the class sizes of a and b in G, respectively. On the other hand, it is also a fact of common knowledge that (normal) subgroups of factorised groups are not necessarily prefactorised (we recall that a subgroup H of a factorised group G = AB is called prefactorised if H = (H ∩ A)(H ∩ B)). Nevertheless, given G = AB, then for each prime p there exists a Sylow p-subgroup P of G such that P = (P ∩ A)(P ∩ B), with P ∩ A and P ∩ B Sylow p-subgroups of A and B, respectively (see [1, 1.3.3]). Moreover, in this situation if either P ∩ A  F(G) or P ∩ B  F(G) (where F(G) denotes the Fitting subgroup of G), then we get that the p-radical subgroup Op (G) is prefactorised. All these key facts appear frequently when proving the results described in this survey. The paper is structured in the following way: in Section 2, we first focus on the case that the class lengths of the elements in the factors of a factorised group are not divisible by p2 , for a fixed prime p, and we also deal with square-free class sizes for all primes. In particular, we consider the case that a given prime p does not divide these conjugacy class sizes. And secondly, in Section 3, we pay attention to the arithmetical condition on the class lengths of being prime powers. We highlight that no permutability condition is assumed in the results in this last section. Moreover, in both cases, we will illustrate the scope of the results presented with some examples, most of them can be found in either [13] or [14]. The notation in the paper is as follows: hereafter G denotes always 9 9a finite group. Let xG be the conjugacy class of an element x ∈ G, and let 9xG 9 be its size. If p is a prime number, we say that x ∈ G is a p-regular element if its order is not divisible by p. The mth group of order n in the SmallGroups library of GAP (see [20]) will be identified by n#m. The remaining notation is standard, and it is taken mostly from [11]. In particular, following this book we write G = [H]K the semidirect product of H and K, where H is normal in G.

2

Square-free class sizes

We begin with the most extreme square-free case: when a given prime p does not divide any class size. This situation was handled by Ballester-Bolinches, Cossey and Li for factorised groups: Theorem 2.1 ([4, Theorem 1.1]) Let the group G = AB be the product of the mutually permutable subgroups A and B, and let p be a prime. Then: 9 9 (1) No conjugacy class length 9xG 9, where x is a p-regular element of prime power order in A ∪ B, is divisible by p if and only if G is p-decomposable, that is,

Felipe et al.: Structural criteria in factorised groups via conjugacy class sizes 291 G = Op (G) × Op (G). 9 9 (2) 9xG 9 is not divisible by p for every element x ∈ A ∪ B if and only if G = Op (G) × Op (G) with Op (G) abelian. Actually, the above theorem is proved in [4] for products of r ≥ 2 subgroups that are pairwise mutually permutable. Chillag and Herzog initially considered groups (not factorised) all of whose class sizes are not divisible by a given prime p in [9, Proposition 4]. This last result can be retrieved from Theorem 2.1 (2) when considering the trivial factorisation G = A = B. Also the first statement of the above theorem extends [16, Theorem 5]. The origin of this strand of research concerning square-free class sizes in factorised groups could be traced in that paper of Liu, Wang and Wei in 2005 ([16]). They proved the following result for products of two permutable subgroups. Recall that a permutable subgroup H of a group G is a subgroup such that HK = KH for every subgroup K of G. Theorem 2.2 ([16, Theorem 10]) 9 9 Let A and B be permutable subgroups of G such that G = AB. Suppose that 9xG 9 is square-free for every element x of A ∪ B. Then G is supersoluble. The above authors in [16, Proposition 9] also obtained the same thesis when the square-free class length hypothesis is imposed on only prime power order elements in the factors, but strengthening the permutability of the factors to normality. (In fact, this last stronger condition is not necessary, as Theorem 2.4 (1) below shows.) If we take the trivial factorisation in Theorem 2.2, then we recover [9, Theorem 1] and [10, Theorem 2], which both analyse groups all of whose conjugacy classes are of square-free length. It is worth remarking that the strong hypotheses in Theorem 2.2 imply that the class length conditions by both factors A and B, that is, if for 9 9 are inherited 9 9 instance a ∈ A, then 9aA 9 divides 9aG 9. This is because both factors are subnormal in G. It is natural then to analyse to what extent the permutability hypothesis on the factors can be relaxed. In this setting, Ballester-Bolinches, Cossey and Li established: Theorem 2.3 ([4, Corollary 1.5]) Let the group G = AB be the product of the mutually permutable subgroups9 A 9and B. Suppose that for every prime p and every p-regular element x ∈ A ∪ B, 9xG 9 is not divisible by p2 . Then G is supersoluble. It is relevant to point out that, in contrast to Theorem 2.2, only p-regular elements in the factors are considered here. A question now arises: does Theorem 2.3 remain true if we apply the hypotheses to only prime power order elements in the factors? The authors of this survey give a positive answer in [14], and also further information on the structure of such groups is provided: Theorem 2.4 Let G = AB be the product of the mutually permutable subgroups A and B. Assume that 9 9for every prime p and every prime power order p-regular element x ∈ A ∪ B, 9xG 9 is not divisible by p2 . Then:

292 Felipe et al.: Structural criteria in factorised groups via conjugacy class sizes (1) ([14, Theorem E]) G is supersoluble, and G/ F(G) has elementary abelian Sylow subgroups. 9 9 (2) ([14, Theorem F]) Assume in addition that p2 does not divide 9xG 9 for all p-regular elements x ∈ A ∪ B, and for every prime p. Then the order of a Sylow p-subgroup of G/ F(G) is bounded by p2 . The above second assertion is not further valid if only prime power order elements in the factors are considered, as the next example shows: Example 2.5 Let G = Sym(3) × D10 × D14 be the direct product of a symmetric group of degree 3 and two dihedral groups of order 10 and 14, respectively. Then G = A × B is a mutually permutable product of A = Sym(3) × D10 and B = D14 , and each prime power order element contained in the direct factors has square-free conjugacy class size. However, G/ F(G) has order 23 . It is worthwhile to note that this construction can be generalised to show that G/ F(G) could be of order 2n for arbitrary n ≥ 3. In [17], Lu, Wei, Zhou and Ma give a similar supersolubility criterion for a product G = AB, but assuming that one of the factors is subnormal in G: Theorem 2.6 ([17]) Let G = AB be the product of the subgroups A and B, and let assume that A is subnormal in G. If for every p and every prime power 9 prime 9 order p-regular element x ∈ A ∪ B it holds that 9xG 9 is not divisible by p2 , then G is supersoluble. Indeed, [10, Theorem 2] due to Cossey and Wang shows that a finite group G such that all of whose conjugacy classes have square-free size has abelian derived subgroup G and |G/ F(G)| is square-free. We have also obtained extensions of these facts as Theorem 2.4 (2) and the next result shows: Theorem 2.7 ([14, Theorem D]) Let G = AB be the product of the subgroups A and B, and assume that G is supersoluble. Suppose that every prime power order element x ∈ A ∪ B has square-free conjugacy class size. Then: (1) G is abelian. (2) The Sylow subgroups of G and G/ F(G) are elementary abelian. (3) F(G) has Sylow p-subgroups of order at most p2 , for every prime p. The supersolubility assumption of G in the previous theorem can be exchanged by the mutual permutability of the factors by Theorem 2.4 (1). Moreover, in relation to the third claim, we also showed that it is not possible to bound the order of G by p2 : Example 2.8 Let G = A × B, where A = D14 is a dihedral group of order 14, and B = D14 × [C7 ]C3 is the direct product of such a dihedral group and a semidirect product of a cyclic group of order 7 and a cyclic group of order 3. Then G is supersoluble, and satisfies that all prime power order elements contained in each factor have square-free conjugacy class size, but G has order 73 .

Felipe et al.: Structural criteria in factorised groups via conjugacy class sizes 293 An example due to Qian and Wang in [18] allows us to point out that Theorem 2.7 (1) is no longer valid under the assumptions in Theorem 2.4, even for arbitrary groups not necessarily factorised: Example 2.9 Let consider a 2-group G = A = B with non-abelian derived subgroup. Then G has a unique 2-regular element, so G clearly satisfies the hypotheses in Theorem 2.4. Motivated by the research performed by Cossey and Wang in [10], we have also handled class lengths of some elements not divisible by p2 , for a fixed prime p, in the context of factorised groups. We have first targeted the problem of p-groups, extending the well-known Knoche’s theorem (see [15]): Theorem 2.10 ([14, Theorem A]) Let 9 9p be a prime number and let P = AB be a p-group such that p2 does not divide 9xP 9 for all x ∈ A ∪ B. Then P   Φ(P )  Z(P ), P  is elementary abelian and |P  |  p2 . We remark that this is the first result within this topic that avoids the use of any permutability property between the factors. Moreover, the converse of the previous theorem is not true (in contrast to what occurs in Knoche’s theorem): Example 2.11 Let P be the group of the SmallGroups library of GAP with identification number 32#35, which is the product of a cyclic group of order 4 and a quaternion group of order 8. Then its derived group has the structure P  = C2 ×C2 , and P  = Φ(P ) = Z(P ). Nevertheless, there are elements in the quaternion group with conjugacy class size in P equal to 4. We have also analysed class sizes not divisible by p2 , for a fixed prime p satisfying (p − 1, |G|) = 1: Theorem 2.12 ([14, Theorem B]) Let G = AB be the product of the mutually permutable subgroups 9 and B, and let p be a prime such that (p − 1, |G|) = 1. If 9 A p2 does not divide 9xG 9 for any p-regular element x ∈ A ∪ B of prime power order, then: (1) G is soluble. (2) G is p-nilpotent. (3) The Sylow p-subgroups of G/ Op (G) are elementary abelian. Note that the assumption that (p−1, |G|) = 1 is always verified by the least prime divisor of |G|, and easy examples show that this condition is necessary. Further, the above result allows us to retrieve the following Qian and Wang’s theorem: Corollary 2.13 ([18, Theorem A]) Let G be a finite group, 9 let p be a prime 9 and such that (p − 1, |G|) = 1. Let assume that p2 does not divide 9xG 9 for any p-regular element x ∈ G of prime power order. Then G is a soluble p-nilpotent group, and the Sylow p-subgroups of G/ Op (G) are elementary abelian.

294 Felipe et al.: Structural criteria in factorised groups via conjugacy class sizes In 2013, Shi, Wei and Ma studied9 products G = AB with A subnormal in G 9 where some elements x ∈ A ∪ B has 9xG 9 not divisible by p2 : Theorem 2.14 ([19, Theorem 5]) Let G = AB be the product of the subgroups A and B, and let suppose that A is subnormal in G. Let p be a prime such 9that9 (p − 1, |G|) = 1. If each prime power order p-regular element x ∈ A ∪ B has 9xG 9 not divisible by p2 , then G is soluble and p-nilpotent. Nevertheless, the third assertion in Theorem 2.12 is not considered in the above result. In addition, note again that 9 in9 Theorem9 2.14 9 the subnormal factor A inherits the class size hypotheses, since 9aA 9 divides 9aG 9 for every a ∈ A. The following example shows that the class size condition might not be acquired by one of the factors under the assumptions of Theorem 2.12. Example 2.15 The group G = [C5 × C5 ](Sym(3) × C2 ) (where Sym(3) is the symmetric group of degree 3) can be factorised as the product of the mutually permutable subgroups A = D10 × D10 and B = [C5 × C5 ]C3 (this can be checked with GAP, [20]). Then G = AB satisfies the hypotheses of Theorem 2.12 9 9for p = 2, but there are 2-regular elements x ∈ A of prime power order with 9xA 9 divisible by 4. This group has the identification 300#25 in the SmallGroups library of GAP. In [4], Ballester-Bolinches, Cossey and Li also obtained: Theorem 2.16 ([4, Theorem 1.3]) Let the group G = AB be the product of the mutually permutable subgroups A and B. Suppose that for every p-regular element 9 9 x ∈ A ∪ B, 9xG 9 is not divisible by p2 . Then the order of a Sylow p-subgroup of every chief factor of G is at most p. In particular, if G is p-soluble, then G is p-supersoluble. We have demonstrated that the second claim is also valid if only prime power order p-regular element x ∈ A ∪ B are considered ([14, Theorem C]). The first assertion of Theorem 2.16 relies heavily on [9, Proposition 3] due to Chillag and Herzog, which uses the classification of finite simple groups, and it asserts the following: if the fixed prime p divides |CG (x)| for all x ∈ G, then G is not a nonabelian simple group. It is an open question whether this result is valid when only prime power order elements are considered. We notice that some results presented in this section have also been collected in [12].

3

Conjugacy classes of prime power size

In this section we meet with prime power class sizes concerning products of groups. It is remarkable that no permutability condition is imposed on the factors. We report mainly on the results in [13], which were motivated by [3], [6] and [7], but focusing in that case on factorised groups. We recall that the structure of groups all of whose conjugacy classes have prime power size was characterised in [3] by Baer. He showed that such groups G have

Felipe et al.: Structural criteria in factorised groups via conjugacy class sizes 295 the following structure: G = G1 × G2 × · · · × Gk , where the direct factors have pairwise coprime orders and, if Gi is not of prime order, then its order is divisible by two primes and its Sylow subgroups are abelian. Indeed, Camina and Camina gave in [7] an alternative shorter proof of Baer’s result, by analysing those groups such that the p-elements have prime power class length, for a fixed prime number p. Such a group is called a p-Baer group in that manuscript. Analogously, it is defined a Baer group as a group such that all of whose prime power order elements have prime power conjugacy class size. Accordingly to this research, we have introduced in [13] the following concepts for a factorised group: Definition 3.1 Let G = AB be the product of the subgroups A and B, and let p be a prime divisor of |G|. Then: 9 9 • G = AB is a p-Baer factorisation if 9xG 9 is a prime power for every p-element x ∈ A ∪ B; 9 9 • G = AB is a Baer factorisation if 9xG 9 is a prime power for all prime power order elements x ∈ A ∪ B, i.e., if it is a p-Baer factorisation for all p. Note that any central product of two (p-)Baer groups provides a (p-)Baer factorisation. The following result due to Camina and Camina is crucial in this development: every element x ∈ G with prime power conjugacy class size lies in the second term of the Fitting series of G (see [7, Theorem 1]). Indeed, this last fact extends a well-known result of Burnside about the non-simplicity of groups with a conjugacy class of prime power size. Besides, Berkovich and Kazarin address several problems about prime power indices in finite groups in [6], and some of their ideas are also useful in the research regarding factorised groups. In particular, in that paper, another alternative shorter proof of Baer’s characterisation is provided. In the next theorem, we obtain structural information of a group G which has a p-Baer factorisation: Theorem 3.2 ([13, Theorem A]) Let G = AB be a p-Baer factorisation, and let P ∈ Sylp (G). Then: (1) G/ CG (Op (G)) is p-decomposable. (2) Both P F(G) and P Op (G) are normal in G. In particular, G is p-soluble of p-length 1. (3) The Sylow p-subgroup of G/ F(G) is abelian. (4) P is abelian if and only if Op (G) is abelian. (5) If P = (P ∩ A)(P ∩ B) and P ∩ X  CG (Op (G)) for some X ∈ {A, B}, then P ∩ X centralises every Hall p -subgroup of G. (6) If the Sylow p-subgroups of A and B are non-abelian, then G is p-decomposable. In fact, we get further information based on the primes appearing as class sizes:

296 Felipe et al.: Structural criteria in factorised groups via conjugacy class sizes Theorem 3.3 ([13, Theorem B]) Let G = AB be a p-Baer factorisation, and 9 G9 9x 9 is a qlet P ∈ Sylp (G). Then there exist unique primes q and r such that 9 9 number for every p-element x ∈ A, and 9y G 9 is an r-number for every p-element y ∈ B, respectively. Moreover, P  CG (O{q,r} (F(G))), and P Oq (G) Or (G) is normal in G. Further: (1) If q = r = p, then G is p-decomposable. (2) If p ∈ / {q, r}, then P is abelian. As a consequence, [7, Theorem A] due to Camina and Camina can be partially obtained, and therefore Baer’s characterisation follows. Corollary 3.4 ([7, Theorem A]) Let G be a p-Baer group. Then: (a) G is p-soluble of p-length 1, and (b) there is a unique prime q such that each p-element has q-power index. Further, let Q ∈ Sylq (G), then (c) if p = q, P is a direct factor of G, or (d) if p = q, P is abelian, and P Oq (G) is normal in G. Corollary 3.5 ([3, Section 3 – Theorem]) Let G be a finite group. Each element x ∈ G of prime power order has prime power index if and only if G = G1 × G 2 × · · · × G r , where Gi and Gj have relatively prime orders for i = j, and if Gi is not of prime order, then |π(Gi )| = 2 and its Sylow subgroups are abelian. Easy examples illustrate that the primes q and r in Theorem 3.3 might be distinct: Example 3.6 Consider the prime p = 2, and let G = A × B where A = Sym(3) and B = D10 . Then G is a 2-Baer factorisation, where the 2-elements in A have 3power index and the 2-elements in B have 5-power index. Since there are 2-elements with index equal to 15, then G it is not a 2-Baer group. Moreover, in relation to Theorem 3.3 (1), it is to be mentioned that in [21] the next extension for a product G = AB with A subnormal in G is obtained: the conjugacy class size of every prime power order π-element in A ∪ B is a π-number if and only if G = Oπ (G) × Oπ (G). As a direct consequence of Theorems 3.2 and 3.3, we get the next result for a Baer factorisation: Corollary 3.7 ([13, Corollary C]) If G = AB is a Baer factorisation, then: (1) G/ F(G) is abelian. (2) G has abelian Sylow subgroups (that is, G is an A-group) if and only if F(G) is abelian.

Felipe et al.: Structural criteria in factorised groups via conjugacy class sizes 297 (3) Set σ := {p ∈ π(G) | Ap ∈ Sylp (A) and Bp ∈ Sylp (B) are non-abelian}. Then G = Oσ (G) × Oσ (G) with Oσ (G) nilpotent. (4) If all Sylow subgroups of A and B are non-abelian, then G is nilpotent. It may be worth asking whether the prime power class length hypotheses are satisfied by the factors of a (p-)Baer factorisation. In contrast to what happens in the previous section, we have established: Proposition 3.8 ([13, Proposition D]) Let G = AB be a Baer factorisation. 9 G9 9 9 Let x ∈ X be a prime power order 9 X 9element, where X ∈ {A, B}. If x is a qnumber for some prime q, then 9x 9 is also a q-number. In particular, it follows that A and B are Baer groups. Therefore the structure of A and B is well-known in a Baer factorisation, by the main theorem in [3]. However, an analogous characterisation as Baer’s one for a Baer factorisation fails to hold, even for direct products G = A × B, as the next example shows: Example 3.9 Consider A = C3 ×[C7 ]C2 ×[C11 ]C5 and B = C5 ×[C7 ]C3 ×[C11 ]C2 . Then G = A × B is a Baer factorisation, but there are no pairwise coprime proper direct factors of G. In relation to Proposition 3.8, we also point out that so far it is not known whether the factors of9 a p-Baer are p-Baer groups. Nevertheless, if for 9 9factorisation 9 instance x ∈ A, then 9xG 9 and 9xA 9 might be powers of distinct primes in a p-Baer factorisation: Example 3.10 Let G be the semidirect product of a non-abelian group of order 21 acting on an elementary abelian group of order 8, in such a way that the subgroup of order 7 permutes the involutions transitively. Then there is a subgroup A of order 24 and a subgroup B of order 7 such that factorisation, 9 G9 = AB is 9a 2-Baer 9 and there exists a 2-element x ∈ A such that 9xA 9 = 3 and 9xG 9 = 7. Additionally, some information arises locally, that is, prime by prime: Theorem 3.11 ([13, Theorem E]) Let G = AB be a Baer factorisation. For a prime p, and given P ∈ Sylp (G): (1) If P is not abelian, then |G : CG (P )| is a {p, q}-number, for a prime q. (2) If P is abelian, then |G : CG (P )| is a {q, r}-number, for some primes q and r, both distinct from p. Further, G/ CG (Op (G)) is p-decomposable with abelian p-complement, and the pcomplement has order divisible by at most two primes. As a consequence, we attain an arithmetical characterisation of Baer factorisations through the indices of the centralisers of the Sylow subgroups of the factors:

298 Felipe et al.: Structural criteria in factorised groups via conjugacy class sizes Theorem 3.12 ([13, Theorem F]) Let G = AB be the product of the subgroups A and B. Then this is a Baer factorisation if and only if |G : CG (Ap )| and |G : CG (Bp )| are prime powers, for Ap ∈ Sylp (A) and Bp ∈ Sylp (B), and for every prime p. Berkovich and Kazarin constructed in [6] the group in the following example, which allows us to illustrate that this last theorem is no longer true for p-Baer factorisations: Example 3.13 Let Q be a cyclic group of order 7. Consider the regular wreath product T = Q wr G where G is the group in Example 3.10, and denote by Q the basis group. Note that the factorisation T = Q G is 2-Baer. If P is the Sylow 2-subgroup of G, then the index |T : CT (P )| is divisible by 3 and 7. Acknowledgements. The first author is supported by Proyecto Prometeo II/ 2015/011, Generalitat Valenciana (Spain). The second author is supported by Proyecto MTM 2014-54707-C3-1-P, Ministerio de Econom´ıa, Industria y Competitividad (Spain), and by Proyecto Prometeo/2017/057, Generalitat Valenciana (Spain). The third author acknowledges the predoctoral grant ACIF/2016/170, Generalitat Valenciana (Spain). References [1] B. Amberg, S. Franciosi and F. de Giovanni, Products of groups, Oxford University Press Inc., (Clarendon Press, New York 1992). [2] M. Asaad and A. Shaalan, On the supersolvability of finite groups, Arch. Math. (Basel) 53 (1989), 318–326. [3] R. Baer, Group elements of prime power index, Trans. Amer. Math. Soc. 75 (1953), 20–47. [4] A. Ballester-Bolinches, J. Cossey and Y. Li, Mutually permutable products and conjugacy classes Monatsh. Math. 170 (2013), 305–310. [5] A. Ballester-Bolinches, R. Esteban-Romero and M. Asaad, Products of finite groups, de Gruyter Expositions in Mathematics 53 (Walter de Gruyter, Berlin 2010). [6] Y. Berkovich and L. S. Kazarin, Indices of elements and normal structure of finite groups, J. Algebra 283 (2005), 564–583. [7] A. R. Camina and R. D. Camina, Implications of conjugacy class size, J. Group Theory 1 3 (1998), 257–269. [8] A. R. Camina and R. D. Camina, The influence of conjugacy class sizes on the structure of finite groups: a survey, Asian-Eur. J. Math. 4 (2011), 559–588. [9] D. Chillag and M. Herzog, On the length of the conjugacy classes of finite groups, J. Algebra 131 (1990), 110–125. [10] J. Cossey and Y. Wang, Remarks on the length of conjugacy classes of finite groups, Comm. Algebra 27 (1999), 4347–4353. [11] K. Doerk and T. Hawkes, Finite Soluble Groups, de Gruyter Expositions in Mathematics 4 (Walter de Gruyter, Berlin 1992). [12] M. J. Felipe, A. Mart´ınez-Pastor and V. M. Ortiz-Sotomayor, On finite groups with square-free conjugacy class sizes, Int. J. Group Theory 7 (2) (2018), 17–24. [13] M. J. Felipe, A. Mart´ınez-Pastor and V. M. Ortiz-Sotomayor, Prime power indices in factorised groups, Mediterr. J. Math. 14 (6), article: 225 (2017).

Felipe et al.: Structural criteria in factorised groups via conjugacy class sizes 299 [14] M. J. Felipe, A. Mart´ınez-Pastor and V. M. Ortiz-Sotomayor, Square-free class sizes in products of groups, J. Algebra 491 (2017), 190–206. ¨ [15] H. G. Knoche, Uber den Frobeniusschen Klassenbegriff in nilpotenten Gruppen, Math. Z. 55 (1951), 71–83. [16] X. Liu, Y. Wang and H. Wei, Notes on the length of conjugacy classes of finite groups, J. Pure Appl. Algebra 196 (2005), 111–117. [17] R. F. Lu, H. Q. Wei, Y. Z. Zhou and X. L. Ma, On a Problem of the Length of Conjugacy Classes of Finite Groups (Chinese), Acta Mathematica Sinica, Chinese series 59 6 (2016), 795–798. [18] G. Qian and Y. Wang, On conjugacy class sizes and character degrees of finite groups, J. Algebra Appl. 13 (2014), 13501001–13501009. [19] X. Shi, H. Wei and X. Ma, Lengths of conjugacy classes of factorized groups and structures of finite groups (Chinese), J. Zhengzhou Univ. Nat. Sci. Ed. 45 2 (2013), 10–13. [20] The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.8.5 (2016), http://www.gap-system.org. [21] X. H. Zhao, X. Y. Guo and J. Y. Shi, On the conjugacy class sizes of prime power order π-elements, Southeast Asian Bull. Math. 35 (2011), 735-740.

GROWTH IN LINEAR ALGEBRAIC GROUPS AND PERMUTATION GROUPS: TOWARDS A UNIFIED PERSPECTIVE HARALD A. HELFGOTT Mathematisches Institut, Georg-August Universit¨at G¨ ottingen, Bunsenstraße 3-5, D-37073 G¨ottingen, Deutschland; IMJ-PRG, UMR 7586, 58 avenue de France, Bˆ atiment S. Germain, case 7012, 75013 Paris CEDEX 13, France Email: [email protected], [email protected]

Abstract By now, we have a product theorem in every finite simple group G of Lie type, with the strength of the bound depending only in the rank of G. Such theorems have numerous consequences: bounds on the diameters of Cayley graphs, spectral gaps, and so forth. For the alternating group Altn , we have a quasipolylogarithmic diameter bound (Helfgott-Seress 2014), but it does not rest on a product theorem. We shall revisit the proof of the bound for Altn , bringing it closer to the proof for linear algebraic groups, and making some common themes clearer. As a result, we will show how to prove a product theorem for Altn – not of full strength, as that would be impossible, but strong enough to imply the diameter bound.

1

Introduction

My personal route in the subject started with the following result. Theorem 1.1 (Product theorem [18]) Let G = SL2 (Z/pZ), p a prime. Let A ⊂ G generate G. Then either |A · A · A| ≥ |A|1+δ or Ak = G, where δ > 0 and k ∈ Z+ are absolute constants.1 Here |S| is the number of elements of a set S, and Ak denotes {a1 . . . ak : ai ∈ A}. (We also write AB for {ab : a ∈ A, b ∈ B} and A−1 for {a−1 : a ∈ A}.) Theorem 1.1 gives us an immediate corollary on the diameter of any Cayley graph Γ(G, A) of G. The diameter of a graph is the maximal distance d(v1 , v2 ) over all pairs of vertices v1 , v2 of a graph Γ; in turn, the distance d(v1 , v2 ) between two vertices is the length of the shortest path between them, where the length of a path is defined as its number of edges. In the particular case of a (directed) Cayley graph Γ(G, A), the diameter equals the least such that every element g ∈ G can be expressed as a product of elements of A of length ≤ . 1

It was soon determined that k = 3 ([17], [6]).

Helfgott: Growth in linear algebraic groups and permutation groups

301

Corollary 1.2 Let G = SL2 (Z/pZ), p a prime. Let S ⊂ G generate G. Then the diameter of the Cayley graph Γ(G, S) is at most (log |G|)C ,

(1.1)

where C is an absolute constant. Proof Apply Theorem 1.1 to A = S, A = S 3 , A = S 9 , etc.



The product theorem has other applications, notably to spectral gaps and expander graphs ([9], [10], [11]). It has been generalized several times, to the point where now it is known to hold for all finite simple2 groups of Lie type and bounded rank [12], [32]. When we say bounded rank, we mean that the constants δ and C in these generalizations of Thm. 1.1 and Cor. 1.2 depend on the rank of the group G. Babai’s conjecture states that the bound (1.1) holds any finite, simple, nonabelian G, and any set of generators S of G, with C an absolute constant (i.e., one constant valid for all G). By the classification of finite simple groups (henceforth: CFSG), every finite, simple, non-abelian group G is either (a) a simple group of Lie type, or (b) an alternating group Alt(n), or (c) one of a finite number of sporadic groups. Being finite in number, the sporadic groups are irrelevant for the purposes of the asymptotic bound (1.1). It remains, then, to consider whether Babai’s conjecture is true for Alt(n), and for simple, finite groups of Lie type whose rank goes to infinity. Part of the problem is that, in either of these two cases, the natural generalization of Thm. 1.1 is false: counterexamples due to Pyber and Spiga [30], [35] show that δ has to depend on the rank of G, or on the index n in Alt(n), at least if there are no additional conditions. Nevertheless, Babai’s conjecture is still believed to be true. Some of the ideas leading to Thm. 1.1 and its generalizations were useful in the proof of the following result, even though the overall argument looked rather different. Theorem 1.3 ([22]) Let G = Alt(n) or G = Sym(n). Then, diam G ≤ eC(log log |G|)

4

log log log |G|

,

(1.2)

where C is an absolute constant. Here we write diam G for the “worst-case diameter” diam G =

max

diam(Γ(G, A)),

A⊂G:G=A

i.e., the same sort of quantity that we bounded in Cor. 1.2. Theorem 1.3 is not as strong as Babai’s conjecture for Alt(n) or Sym(n), since the quantity on the right of (1.2) is larger than (log |G|)C . The proof of Thm. 1.3 2 It is trivial to see that the theorem and corollary above still hold if SL2 (Z/pZ) is replaced by the simple group PSL2 (Z/pZ).

302

Helfgott: Growth in linear algebraic groups and permutation groups

did not go through the proof of an analogue of a product theorem; it used another kind of inductive process. One of our main aims in what follows is to give a different proof of Theorem 1.3. Some of its elements are essentially the same as in the original proof, sometimes in improved or simplified versions. Others are more closely inspired by the tools developed for the case of groups of Lie type. Theorem 1.3 – or rather a marginally weaker version thereof (Thm. 6.1), with an additional factor of log log log |G| in the exponent – will follow as a direct consequence of the following product theorem, which is new. It is, naturally, weaker than a literal analogue of Thm. 1.1, since such as analogue would be false, by the counterexamples we mentioned. Theorem 1.4 Let A ⊂ Sym(n) be such that A = A−1 , e ∈ A, and A is 3transitive. There are absolute constants C, c > 0 such that the following holds. 2 Assume that |A| ≥ nC(log n) . Then either C

|An | ≥ |A|

1+c

log log |A| (log n)2 log log n

(1.3)

or diam(Γ(A , A)) ≤ nC diam(G),

(1.4)

where G is a transitive group on m ≤ n elements such that either (a) m ≤ e−1/10 n or (b) G  Alt(m), Sym(m). Our general objective will be to make the proof for Alt(n) and Sym(n) not just simpler but closer to that for groups of Lie type of bounded rank. Part of the motivation is that the next natural aim is to study in depth groups of Lie type of unbounded rank, which combine features of both kinds of groups. Overall idea. To prove the growth of sets A in a group G, we study the actions of a group G. First of all, every group acts on itself, by left and right multiplication, and by conjugation. The study of these actions is always useful; it gives us lemmas valid for every group. Then there are the actions that exist for a given kind of group. A linear algebraic group acts by linear transformations on affine space. It then makes sense to see how the action of the group affects varieties, and what this tells us about sets of elements in the group. In the case of the symmetric group Sym(Ω), |Ω| = n, we have no such nicely geometric action. What we do have is an action on a set Ω, that, while completely unstructured, is very small compared to the group. This fact allows us to use short random walks to obtain elements whose action on Ω and low powers follows an almost uniform distribution. It is then unsurprising that the strategies for linear algebraic groups and symmetric groups diverge: the actions that characterize the two kinds differ. Nevertheless, it is possible to unify the strategies to some extent. We shall see that the role played by generic elements – in the sense of algebraic geometry – in the study of growth in linear algebraic groups is roughly analogous to the role played in permutation groups by random elements – in the sense of being produced by random walks.

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303

Further perspectives. A “purer” product theorem would state that either (1.3) C log n holds or, say, An = G. The switch to diameters in conclusion (1.4) is not just somewhat ungainly; it also slows down the recursion. If (1.4) were replaced by C log n = G, we would then obtain an exponent of 3 instead of 4 in Theorem 1.3. An Such a “purer” result is not contradicted by the existing counterexamples, and so remains a plausible goal. Yet another worthwhile goal would be to remove the dependence on the Classification of Finite Simple Groups (CFSG). The proof here uses the structure theorem in [13]/[25], which relies on CFSG. The proof in [22] also depended on CFSG, for essentially the same reason: it used [8, Thm. 1.4], which uses [13]/[25]. Incidentally, there is a flaw in [8, Thm. 1.4] (proof and statement), as L. Pyber pointed out to the author. We fix it in §4 (with input from Pyber); the amended statement is in Prop. 4.16. The bound in [22] is not affected when we replace [8, Thm. 1.4] by Prop. 4.16 in the proof of [22]. Notation. We write actions on the right, i.e., if G acts on X, and g ∈ G, x ∈ X, we write xg for the element to which g sends x. As is usual, we write f (x) = O(g(x)) to mean that there exists a constant C > 0 (called an implied constant) such that |f (x)| ≤ Cg(x) for all large enough x. We also write f (x) ' g(x) to mean that f (x) = O(g(x)), and f (x) ( g(x) to mean, for g taking positive values, that there is a constant c > 0 (called, again, an implied constant) such that f (x) ≥ cg(x) for all large enough x. When we write O∗ (c), c a non-negative real, we simply mean a quantity whose absolute value is at most c. Given h ∈ G, we write C(h) for the centralizer {g ∈ G : gh = hg} of h. Given H ≤ G, we write C(H) for the centralizer {g ∈ G : gh = hg ∀h ∈ H} of H. As should be clear by now, and as is standard, we write Alt(Ω) for the alternating group on a set Ω, and Alt(m) for the abstract group isomorphic to Alt(Ω) for any set Ω with n elements. We define [n] = {1, 2, . . . , n}. Acknowledgements. The author is supported by ERC Consolidator grant GRANT 648329, and by funds from his Humboldt professorship. He is deeply grateful to L´ aszl´o Pyber for his extremely valuable suggestions and feedback, and to Henry Bradford and Vladimir Finkelshtein, for a very careful reading and many corrections.

2 2.1

Toolbox Special sets

In the proofs of growth for groups of Lie type, some of the main tools are statements on intersections with varieties. A typical statement is of the following kind. Lemma 2.1 Let G = SL2 (K), K a finite field. Let A ⊂ G be a set of generators of G with A = A−1 . Let V be a one-dimensional irreducible subvariety of SL2 . Then, for every δ > 0, either |A3 | ≥ |A|1+δ holds, or the intersection of A with V has 1 +O(δ) = |A|1/3+O(δ) ' |A| dim SL2

304

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elements. The implied constants depend only on the degree of V . Special statements of this kind were proved and used in [18] and [19], and have been central to the main strategy since then. They were fully generalized in [32]. As it happens, Larsen and Pink, in the course of their work on finite subgroups of linear groups, had proven results of the same kind – for subgroups H, instead of sets A, but for all simple linear groups G. Their procedure was adapted in [12] to give essentially the same general result as in [32]. (Incidentally, the main purpose of Larsen and Pink was to prove without CFSG a series of statements that follows from CFSG. For this purpose, they developed tools that were, in some sense, both concrete and general. It was these features that let the tools be generalized later to sets, as opposed to subgroups. This is not the only time that preexistent work on doing without CFSG has proved fruitful in this context; we will see another instance when we examine random walks and permutation groups.) There is an obvious difficulty in adapting such work to the study of permutation groups: in Sym(n), there seems to be no natural concept of a “variety”, let alone of its degree and its dimension. The approach we will follow here is to strip to the proof of a statement such as Lemma 2.1 to its barest bones, so that the main idea becomes a statement about an abstract group. We will later be able to see how to apply it to obtain a useful result on permutation groups. The proof of Lemma 2.1 goes as follows. First, we show that, for generic g1 , g2 ∈ SL2 (K), the map φ : V × V × V → G given by φ(v0 , v1 , v2 ) = v0 · g1 V g1−1 · g2 V g2−1 is almost-injective, in the sense that the preimage of a generic point of the (closure of) the image is zero-dimensional. “A generic point” here means “a point outside a subvariety of positive codimension”. Similarly, “for g1 , g2 generic” means that the pairs (g1 , g2 ) for which the map φ is not almost-injective lie in a variety of positive codimension W in SL2 × SL2 . Now, because A generates G, a general statement on escape of subvarieties shows that there exists a pair (g1 , g2 ) ∈ Ak × Ak outside W , where k is a constant depending only on the degree of V . (“Escape of subvarieties” was an argument known before [18]. The statement in [15, Prop. 3.2] is over C, but the argument of the proof there is valid over an arbitrary field; see, e.g., [19, Prop. 4.1].) Then we examine the image of (A ∩ V ) × (A ∩ V ) × (A ∩ V ) under φ. If φ is injective, then the image has exactly the same size as the domain, namely, |A∩V |3 . In general, for φ almost-injective, the image will have size ( |A ∩ V |3 . At the same time, the image is contained in A1+2k+1+2k+2k+1+2k = A8k+3 . Hence 91/3 9 9 9 |A ∩ V | ≤ 9A8k+3 9 . Let us prove an extremely simple general statement that expresses the main idea of the statement we have just sketched.

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305

Lemma 2.2 Let G be a group. Let A, B ⊂ G be finite. Then |AB −1 | ≥

|A||B| . |AA−1 ∩ BB −1 |

In particular, if AA−1 ∩ BB −1 = {e}, then |AB −1 | ≥ |A||B|. The condition AA−1 ∩ BB −1 = {e} is fulfilled if, for instance, A ⊂ H1 , B ⊂ H2 , where H1 , H2 are subgroups of G with H1 ∩ H2 = {e}. Proof Consider the map φ : A × B → AB −1 ⊂ G defined by (a, b) → ab−1 . Clearly, as with any map from A × B to G, | im(φ)| ≥

|A × B| , maxx∈G |φ−1 (x)|

(2.1)

and of course |AB −1 | ≥ | im(φ)|. So, let us bound φ−1 (x). Say φ(a, b) = x = φ(a , b ). Then a−1 a = b(b )−1 .

(2.2)

In particular, given a, b and b(b )−1 , we can reconstruct a and b . Moreover, again by (2.2), b(b )−1 lies in AA−1 ∩ BB −1 . Letting (a, b) be fixed, and letting (a , b ) vary among all elements of φ−1 (x), we see that 9 −1 9 9φ (x)9 ≤ |AA−1 ∩ BB −1 |. 

By (2.1), we are done.

We can apply the same idea to obtain growth assuming only that an intersection of many sets is empty. Lemma 2.3 Let G be a group. Let A0 , A1 , . . . , Ak ⊂ G be finite. Then there is at least one 0 ≤ j ≤ k − 1 such that k+1 9 9 A k 9 9 ≥ 9Aj A−1 9 9 91/k , j+1 9 k 9 9 j=0 Aj A−1 j 9

where A is the geometric average (

k

k

j=0 |Aj |)

1/(k+1) .

Aj A−1 j = {e},

j=0

then

9 9 k+1 9 9 k . 9Aj A−1 j+1 9 ≥ A

In particular, if (2.3)

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We will typically apply this lemma to sets Aj that are conjugates of each other,  and so all of the same size A. If A ⊂ H, Aj = gj Agj−1 and kj=0 gj Hg −1 = {e}, then condition (2.3) holds. Proof Consider the map −1 −1 φ : A0 × A1 × . . . × Ak → A0 A−1 1 × A1 A2 × . . . × Ak−1 Ak

given by

−1 −1 (a0 , a1 , . . . , ak ) → a0 a−1 . 1 , a1 a2 , . . . , ak−1 ak

Clearly, k−1 

k

|Aj A−1 j+1 | = | im(φ)| ≥

j=0

j=0 |Aj | . maxx∈G |φ−1 (x)|

: ;−1   Say φ(a0 , a1 , . . . , aj ) = x = φ(a0 , a1 , . . . , aj ). Then, since aj a−1 j+1 = aj aj+1 −1   for all 0 ≤ j < k, we see that a−1 j aj = aj+1 aj+1 for all 0 ≤ j < k. Thus, (a0 , a1 , . . . , aj ) is determined by (a0 , a1 , . . . , aj ) and the single element −1  −1   a−1 0 a0 = a1 a1 = . . . = ak ak ,

which lies in

k

−1 j=0 Aj Aj .

We conclude that 9 9 9 k 9 9 9 −1 9 −1 9 |φ (x)| ≤ 9 Aj Aj 9 . 9j=0 9 

We will later see how to obtain the weak orthogonality condition k

gj Hgj−1 = {e}

(2.4)

j=0

for some kinds of subgroups of permutation groups. 2.2

Subgroups and quotients

We will need a couple of basic lemmas on subgroups and quotients. As explained in [22, §3.1–3.2] and [20, §4.1], they are all easy applications of an orbit-stabilizer principle for sets [20, Lemma 4.1]. We can also prove them by using the pigeonhole principle directly. For G a group and H ≤ G, we write πG/H : G → G/H for the map taking each g ∈ G to the right coset Hg containing g. Thus, for instance, |πG/H (A)| equals the number of distinct cosets Hg intersecting A.

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307

Lemma 2.4 [[19, Lem. 7.2]] Let G be a group and H a subgroup thereof. Let A ⊆ G be a non-empty finite set. Then |AA−1 ∩ H| ≥

|A| |A| ≥ . |πG/H (A)| [G : H]

(2.5)

Proof By pigeonhole, there is a coset Hg of H containing at least |A|/|πG/H (A)| elements of A. Fix g0 ∈ A ∩ Hg. Then, for each g1 ∈ A ∩ Hg, we obtain a distinct element g0 g1−1 ∈ AA−1 ∩ H.  Lemma 2.5 Let G be a group and H a subgroup thereof. Let A ⊆ G be a nonempty finite set. Then, for any k ≥ 1, 9 k 9 9 9 9A ∩ H 9 9 k+1 9 · |A|. 9A 9 ≥ |AA−1 ∩ H|

(2.6)

In other words, growth in a subgroup implies growth in the group. Proof It is clear that 9 9 9: 9 9 ; 9 9 9 9 k+1 9 9 k 9 9 k9 9A 9 ≥ 9 A ∩ H · A9 ≥ 9(A ∩ H) 9 · 9πG/H (A)9 . At the same time, by Lemma 2.4, 9 9

9 9 9 AA−1 ∩ H 9 · 9πG/H (A)9 ≥ |A|.  Lemma 2.6 [[22, Lem. 3.7]] Let G be a group, let H, K be subgroups of G with H ≤ K, and let A ⊆ G be a non-empty finite set. Then |πK/H (AA−1 ∩ K)| ≥

|πG/H (A)| |πG/H (A)| ≥ . |πG/K (A)| [G : K]

In other words: if A intersects r[G : H] cosets of H in G, then AA−1 intersects at least r[G : H]/[G : K] = r[K : H] cosets of H in K. (As usual, all of our cosets are right cosets Hg, Kg, etc.) We quote the proof in [22, Lem. 3.7]. Proof Since A intersects |πG/H (A)| cosets of H in G and |πG/K (A)| cosets of K in G, and every coset of K in G is a disjoint union of cosets of H in G, the pigeonhole principle implies that there exists a coset Kg of K such that A intersects at least k = |πG/H (A)|/|πG/K (A)| cosets Ha ⊆ Kg. Let a1 , . . . , ak be elements of A in −1 ∩ K for each i = 1, . . . , k. Note distinct cosets of H in Kg. Then ai a−1 1 ∈ AA −1 −1 that Ha1 a1 , . . . , Hak a1 are k distinct cosets of H. 

308 2.3

Helfgott: Growth in linear algebraic groups and permutation groups Graphs and random walks

For us, a graph is a directed graph, that is, a pair (V, E), where V is a set and E is a subset of the set of ordered pairs of elements of V . (We allow loops, that is pairs, (v, v).) A multigraph is the same as a graph, but with E a multiset, i.e., edges may have multiplicity > 1. Given a group G and a set of generators A ⊂ G, the Cayley graph Γ(G, A) is defined to be the pair (G, {(g, ga) : g ∈ G, a ∈ A}). It is connected because A is a set of generators. Given a group G, a set of generators A ⊂ G and a set X on which G acts, the Schreier graph Γ(G, A; X) is the pair (X, {(x, xa ) : x ∈ X, a ∈ A}). We take a random walk on a graph or multigraph Γ by starting at a given vertex v0 and deciding randomly, at each step, to which neighbor w of our current location v to move. (A neighbor of v is a vertex w such that (v, w) ∈ E.) We choose w with uniform probability among the neighbors of v, if Γ is a graph, or with probability proportional to the multiplicity of w, if Γ is a multigraph. In a lazy random walk, at each step, we first throw a fair coin to decide whether we are going to bother to move at all. (Of course, if we decide to move, and (v, v) is an edge, we might move from v to itself.) Our random walks will always be lazy, for the sake of eliminating some technicalities. We say that the ( ∞ , )-mixing time in a regular, symmetric (multi)graph Γ = (V, E) is at most t if, for every (lazy) random walk of length ≥ t, the probability that it ends at any given vertex lies between (1 − )/|V | and (1 + )/|V |. We will use the fact that (multi)graphs with few vertices have small mixing times. Proposition 2.7 Let Γ be a connected, regular and symmetric multigraph of valency d and with N vertices. Then the ( ∞ , )-mixing time is at most N 2 d log(N/). Proof This is a well-known fact; see, e.g., the exposition in [20, §6]. The main idea is to study the spectrum of the adjacency operator, meaning the operator A taking each function f : V → C to a function A f whose value at v is the average of f (w) over the neighbors w of v in the graph Γ. The connectedness of Γ is used to show that, for every non-constant eigenfunction of A , the corresponding eigenvalue λ cannot be too close to 1; it is at most 1 − 1/N 2 d. The bound on the mixing time then follows.  In particular, Prop. 2.7 holds when Γ is any Schreier graph Γ(G, A; Ω(k) ) of the (k) action of a permutation group G  Sym(Ω) 9 9on the set Ω of k-tuples of distinct elements of Ω. The point is that N = 9Ω(k) 9 ≤ |Ω|k /k! is very small compared to Sym(Ω) (which is of course of size |Ω|!) for k bounded. We can make sure that A is small as well, by the following simple lemma. Lemma 2.8 Let A ⊂ Sym(n). Then there is a subset A0 ⊂ A ∪ A−1 such that A0 = A , |A0 | ≤ 4n and A0 = A−1 0 . Proof Choose an element g1 ∈ A, and then an element g2 ∈ A such that g1  g1 , g2 , and then an element g3 ∈ A such that g1 , g2  g1 , g2 , g3 , . . . Since the longest subgroup chain in Sym(n) is of length ≤ 2n − 3 [2], we must stop in r ≤ 2n − 3 < 2n steps. Let A0 = {g1 , g1−1 , . . . , gr , gr−1 }. 

Helfgott: Growth in linear algebraic groups and permutation groups

309

The point is that, while we cannot assume we can produce random, uniformly distributed elements of G = A as short products in A (we cannot assume what we are trying to prove, namely, that the diameter is small), we can take short, random products of elements of A, and their action on Ω(k) is like that of random, uniformly distributed elements. This observation was already used in [4] to prove the following. Lemma 2.9 ([4]) Let A ⊂ Sym(Ω), |Ω| = n, be such that A = A−1 and G = A is 3-transitive. Assume there is a g ∈ A, g = e, with | supp(g)| ≤ (1/3 − )n,  > 0. Then diam(Γ(G, A)) ' n8 (log n)c , where c is an absolute constant. Proof [Sketch of proof] See [4] or the exposition in [20, §6.2]. The main idea is as m be the outcome of a follows. Let A0 be as in Lemma 2.8, and let h ∈ Am 0 ⊂ A 3  random walk on A0 of length ≤ m, where m ≥ 4n log(n/ ) and  = /100 (say). Then, by Prop. 2.7, for any x, y ∈ Ω, the probability that h takes x to y is almost exactly 1/n. In particular, for x ∈ supp(g), the probability that xh ∈ supp(g) is almost exactly | supp(g)|/n. Hence 2 9

9 9supp(g) ∩ supp hgh−1 9  | supp(g)| ≤ n



 1 −  | supp(g)|. 3

A quick calculation shows that the commutator [g, h−1 ] = g −1 hgh−1 obeys 9 < =

9 | supp g, h−1 | ≤ 3 9supp(g) ∩ supp hgh−1 9 , = < and so, for g  = g, h−1 , | supp(g  )| ≤ | supp(g)|2 /n ≤ (1 − 3)| supp(g)|. We iterate until, after O(log log n), we obtain an element h of support of size 2 or 3. (Additional care is taken in the process so that our element h is never trivial. It is, in fact, convenient to take m ≥ 4n5 log(n/ ) from the beginning, so that the probability that h takes a pair (x, x ) ∈ Ω(2) to a pair (y, y  ) ∈ Ω(2) is almost 3 exactly 1/|Ω|(2) = 2/n(n − 1).) We conjugate h by elements of An0 to obtain a set C consisting of all 2-cycles or 3-cycles. It is clear that diam(Γ(G, C)) ' n.  We shall now use short random walks to construct elements gj such that a weak orthogonality condition in the sense of (2.4) holds for some kinds of sets B. By an orbit of a set B ⊂ Sym(Ω) we mean a subset of Ω of the form xB , x ∈ Ω. Lemma 2.10 Let A, B ⊂ Sym(Ω), |Ω| = n, e ∈ B. Let 0 < ρ < 1. Assume that A is 2-transitive, and that

m B has no orbits of length > ρn. Then there are g1 , g2 , . . . , gk ∈ A ∪ A−1 ∪ {e} , k ' (log n)/| log ρ|, m ' n6 log n, such that k j=1

gj Bgj−1 = {e}.

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Helfgott: Growth in linear algebraic groups and permutation groups

If we required only that g ∈ A (and not gk ∈ (A ∪ A−1 ∪ {e})m ), and B were assumed to be a group, then this Lemma would be the “splitting lemma” in [1, §3]. The fact that the proof can be adapted illustrates what we were saying: short random products act on Ω(2) as random elements do. Prop. 5.2 in [22] is an earlier generalization of Babai’s “splitting Lemma”, based on the same idea. Proof Let g1 , . . . , gk be the outcome of k independent random walks of length ≤ m on A0 , where m ≥ 4n6 log(n/),  > 0 and A0 ⊂ A ∪ A−1 is as in Lemma 2.8. Then, by Prop. 2.7, for any (x, y), (x , y  ) ∈ Ω(2) and9 any9 1 ≤ j ≤ k, the9 probability 9 that gj takes (x, y) to (x , y  ) lies between (1 − )/ 9Ω(2) 9 and (1 + )/ 9Ω(2) 9. 9 9 Since B has no orbits of length > ρn, there are at most ρ 9Ω(2) 9 pairs (x , y  ) ∈ Ω(2) such that x and y  lie in the same orbit of H. Hence, for any (x, y) ∈ Ω(2) and any 1 ≤ j ≤ k, the probability that xgj and y gj lie in the same orbit of B is at most (1 + )ρ. Since g1 , . . . , gk were chosen independently, it follows that the probability that xgj and y gj are in the same orbit for every 1 ≤ j ≤ k is at most ((1 + )ρ)k . Now, xgj and y gj are in the same  orbit for every 1 ≤ j ≤ k if and only if x and y are in the same orbit of B  = kj=1 gj Bgj−1 . The probability that at least two distinct x, y lie in the same orbit of B  is therefore at most n2 ((1 + )ρ)m . We let3  = ρ−1/2 − 1, so that ((1 + )ρ) = ρ1/2 . Then, for k > 2(log n2 )/| log ρ|, n2 ((1 + )ρ)m < 1. In other words, with positive probability, no two distinct x, y lie in the same orbit of B  , i.e., B  equals {e}. Thus, there exist g1 , . . . , gm such that B  = {e}.  It is a familiar procedure in combinatorics (sometimes called the probabilistic method) to prove that a lion can be found at a random place of the city with positive probability, and to conclude that there must be a lion in the city. What we have done is prove that, after a short random walk, we come across a lion with positive probability (and so there is a lion in the city). Corollary 2.11 Let A, B ⊂ Sym(Ω), |Ω| = n. Let 0 < ρ < 1. Assume that A is 2-transitive, and that BB −1 has no orbits of length > ρn. Then there is a m g ∈ A ∪ A−1 ∪ {e} , m ' n6 log n, such that | log ρ| 9 9 9BB −1 gBB −1 g −1 9 ≥ |B|1+ log n .

−1 Proof By

mLemma 2.10 applied to BB 6 rather than B, there are g1 , g2 , . . . , gk ∈ −1 A∪A , k ' (log n)/| log ρ|, m ' n log n, such that k

gj BB −1 gj−1 = {e}.

j=1 3

We can assume ρ ≤ (n − 1)/n. Thus  = ρ−1/2 − 1 implies   1/n, and so log(n/) log n.

Helfgott: Growth in linear algebraic groups and permutation groups

311

Hence, by Lemma 2.3, with Aj = gj BB −1 gj−1 (and g0 = e, say), there is a 0 ≤ j ≤ k − 1 such that 9 9 9 9 1 1 9 9 9 −1 91+ k |B|1+ k . 9Aj A−1 j+1 9 ≥ BB −1 −1 −1 −1 −1 −1 −1 −1 Now |Aj A−1 j+1 | = |gj BB gj gj+1 BB gj+1 | = |BB gBB g | for g = gj gj+1 , so we are done. 

Corollary 2.12 Let A ⊂ Sym(Ω), |Ω| = n, with A = A−1 and e ∈ A. Let 0 < ρ < 1. Assume that A is 2-transitive. Let Σ ⊂ Ω be such that (A4 )(Σ) has no orbits of length > ρn. Then either |Σ| ≥ or

| log ρ| log |A| 3(log n)2

(2.7)

9 9 | log ρ| 9 l9 1+ 9A 9 ≥ |A| 3 log n

(2.8)

for some l ' n6 log n. Compare to [22, Cor. 5.3]. Proof Let B = (A2 )(Σ) = (AA−1 )(Σ) . Since BB −1 ⊂ (A4 )Σ , we see that BB −1 has no orbits of length > ρn. Apply Corollary 2.11. We obtain that 9 9 | log ρ| 9 l9 9A 9 ≥ |B|1+ n for l = 4m + 2, m ' n6 log n. At the same time, by Lemma 2.4, 9 9 |B| = 9AA−1 ∩ Sym(Ω)(Σ) 9 ≥

|A| |A| ≥ |Σ| . [Sym(Ω) : Sym(Ω)(Σ) ] n

Hence, either (2.7) holds, or |A|

|B| > n and so

| log ρ| 2(log n)2

log |A|

= |A|

| log ρ|

1− 2 log n

,

     9 9 | log ρ| | log ρ| | log ρ| 1− 2 log n 1+ log n 1− log1 n 1+ 9 l9 = |A| 2 log n . 9A 9 ≥ |A|

We can assume 1 − 1/ log n ≥ 2/3, as otherwise (2.8) holds trivially. 2.4



Generating an element of large support

We will need to produce an element of Sym(n) of very large support (almost all of {1, 2, . . . , n}). It is not difficult to carry out this task using short random walks.

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Helfgott: Growth in linear algebraic groups and permutation groups

Lemma 2.13 Let g ∈ Sym(n) have support ≥ αn, α > 0. Let A ⊂ Sym(n) generate a 2-transitive group. Assume A = A−1 , e ∈ A. Then, provided that n is 6 larger than a constant depending only on α, there are γi ∈ An , 1 ≤ i ≤ , where

= O((log n)/α), such that the support of γ1 gγ1−1 · γ2 gγ2−1 · · · γ gγ−1 has at least n − 1 elements. Proof Let h1 , h2 ∈ Sym(n), mi = | supp(hi )|. By Prop. 2.7, a random walk of length r = )4n5 log(n2 /)* gives us an element σ of Ar sending any given pair of distinct elements x, y ∈ {1, . . . , n} to any given pair of distinct elements x , y  ∈ {1, . . . , n} with probability (1 + O∗ ())/n(n − 1). An element x ∈ {1, . . . , n} can fail to be in the support of h1 σh2 σ −1 only if (a) x∈ / supp(h1 ), x ∈ / supp σh2 σ −1 , or (b) x ∈ supp(h1 ) and σh2 σ −1 sends xh1 to x. For x random, case (a) happens with probability at most (1 − m1 /n) · (1 + )(1 − m2 /n). In case (b), σ must send xh1 to an element that is not fixed by h2 , and, moreover, it must send x to xh1 σh2 . Now, we know that, even given that σ sends an element x0 (in this case, x0 = xh1 ) to some specific element y0 , it will still send any x = x0 to any y = y0 with almost equal probability. Hence,  : m2 ; m1 m2 1 m1 ; : Prob(x ∈ / supp(h1 σh2 σ −1 )) ≤ (1 + ) 1− + 1− n n n n n−1 We set  = 1/n and assume m1 , m2 < n. Then we have : m2 ; 1 m1 ; : 1− + . Prob(x ∈ / supp(h1 σh2 σ −1 )) ≤ (1 + ) 1 − n n n The expected value of n − | supp(h1 σh2 σ −1 )| is thus at least (1 + )(n − m1 )(1 − m2 /n) + 1/n. Hence there is a σ ∈ Ar such that n − | supp(h1 σh2 σ −1 )| is at least that much. We apply this first with h1 = h2 = g, and obtain a σ1 = σ as above; define g1 = gσ1 gσ1−1 . Then we iterate: we let h1 = g1 , h2 = g, and obtain a σ2 = σ such that g2 = g1 σ2 gσ2−1 has large support; and so forth, with h1 = gi−1 , h2 = g at the ith step. We obtain    supp(gi ) supp(g) supp(gi−1 ) 1 1− ≥ (1 + ) 1 − 1− + , n n n n where g0 = g, and so, for r = (1 + )(1 − supp(g)/n) (which is < 1) and k ≥ 0,   supp(gk ) supp(g) 1 1 k 1− ≥r 1− + ≥ rk (1 − α) + . n n (1 − r)n (1 − r)n We let k = )(log n)/(log 1/r)* and obtain supp(gk ) ≥ n − 1 −

1 . (1 − r)

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313

For n ≥ 2/α, we have r ≤ (1 + α/2)(1 − α) < 1 − α/2, and so 1/(1 − r) ≤ 2/α and k ' (log n)/α. We can assume supp(gk ) < n, as otherwise we are done. Now apply the procedure at the beginning with h1 = h2 = gk . We obtain Prob(x ∈ / supp(gk σgk σ −1 )) < 2/n, provided that n is larger than a constant depending only on α. Hence there is a σ ∈ Ar such that supp(gk σgk σ −1 ) ≥ n − 1. Since gk σgk σ −1 = g · σ1 gσ1−1 · · · σk gσk−1 · σgσ −1 · (σσ1 )g(σσ1 )−1 · · · (σσk )g(σσk )−1 , 

we set = 2k + 2 and are done.

It may be useful to compare Lemma 2.13 to analogous results on random subproducts in the sense of [5]. Such results make weaker assumptions (transitivity instead of double transitivity) and give weaker conclusions (support ≥ n/2 instead of support ∼ n; see [33, Lemma 2.3.1], [22, Lemma 4.3]). 2.5

Stabilizers and stabilizer chains

Let A ⊂ Sym(Ω), |Ω| = n. Given a subset Σ = {α1 , α2 , . . . , αk } ⊂ Ω, we write A(Σ) and AΣ for the pointwise and setwise stabilizers, respectively: A(Σ) = A(α1 ,...,αk ) = {g ∈ A : αjg = αj ∀1 ≤ j ≤ k}, AΣ = A{α1 ,...,αk } = {g ∈ A : Σg = Σ}. A stabilizer chain is simply a chain of subsets A ⊃ A(α1 ) ⊃ A(α1 ,α2 ) ⊃ . . . , where α1 , α2 , . . . ∈ {1, 2, . . . , n}. Stabilizer chains have been studied starting with Sims [34] (in the case of A equal to a subgroup H). It is useful to find long chains of stabilizers such that the orbits A(α1 ,...,αj−1 )

αj

are long. Why do we want stabilizer chains with long orbits? Here is one reason. Lemma 2.14 Let A ⊂ Sym(Ω), |Ω| = n. Let ρ ∈ (0, 1). Let Σ = {α1 , . . . , αk } ⊂ Ω be such that, for every 1 ≤ j ≤ k, 9 9 9 A(α1 ,...,αj−1 ) 9 9α 9 ≥ ρn. (2.9) 9 j 9 Then Ak intersects at least (ρn)k right cosets of Sym(Ω)(Σ) , and the restriction of the setwise stabiliser (A−k Ak )Σ to Σ is a subset of Sym(Σ) with at least ρk k! elements. This result was shown in the proof of [31, Lemma 3] for A a subgroup, and in [22, Lemma 3.19] for general A.

314

Helfgott: Growth in linear algebraic groups and permutation groups

Proof First of all, notice that Ak sends (α1 , α2 , . . . , αk ) to at least (ρn)k distinct k-tuples. This is shown as follows. Let 1 ≤ j ≤ k. Let Δj denote the orbit A(α

,...,α

)

j−1 αj 1 . For each δ ∈ Δj , choose an element gδ ∈ A(α1 ,...,αj−1 ) sending αj to δ. Let Si = {gδ : δ ∈ Δi }. Clearly, |Si | = |Δi | and Si ⊂ A. Now let (s1 , s2 , . . . , sk ), (s1 , s2 , . . . , sk ) be two distinct elements of S1 × · · · × Sk . Then sk · · · s2 s1 and sk · · · s2 s1 send (α1 , α2 , . . . , αk ) to two different k-tuples: if j is the least index such that sj = sj , then

s sk−1 ···sj

αj k and so

s sk−1 ···sj sj−1 ···s1

αj k

s

s

s sk−1 ···sj

= αj j = αj j = αj k s sk−1 ···sj sj−1 ···s1

= αj k

,

s sk−1 ···sj sj−1 ···s1

= αj k

.

Hence (α1 , α2 , . . . , αk ) is sent to at least |S1 | · · · |Sk | ≥ (ρn)k distinct tuples by the action of Sk Sk−1 · · · S1 ⊂ Ak . In other words, Ak intersects at ≥ (ρn)k cosets (Sym(Ω))(Σ) g of (Sym(Ω))(Σ) . By Lemma 2.6, π(Sym(Ω))Σ /(Sym(Ω))(Σ) (Ak A−k ∩ (Sym(Ω))Σ ) ≥

πSym(Ω)//(Sym(Ω))(Σ) (Ak )

[Sym(Ω) : (Sym(Ω))Σ ] (ρn)k ≥ ≥ ρk k!. n(n − 1) . . . (n − k + 1)/k!

Now, two elements of (Sym(Ω))Σ lie in different cosets of (Sym(Ω))(Σ) if and only their restrictions to Σ are distinct. Since Ak A−k ∩ (Sym(Ω))Σ = (Ak A−k )Σ , we have shown that the restriction (A−k Ak )Σ to Σ is of size at least ρk k!.  2.6

Composition factors. Primitive groups

Let us recall some standard definitions. A composition factor of a group G is a quotient Hi+1 /Hi in a composition series of G, i.e., a series 1 = H0  H1  . . .  Hn = G, where every quotient is simple. By the Jordan-H¨older theorem, whether or not an abstract group is a composition factor of G does not depend on the particular composition series of G being used. A section of a group G is a quotient H/N , where 1 ≤ N  H ≤ G. A composition factor is, by definition, a section. A block system for a permutation group G ≤ Sym(Ω) is a partition of Ω preserved by G, that is, a partition of Ω into blocks (sets) B1 , . . . , Bk such that, if x, y ∈ Bi , g ∈ G and xg ∈ Bj , then y g ∈ Bj . A maximal block system is one that has blocks of size > 1 and cannot be subdivided into a finer partition with blocks of size > 1. (In other words, it is a system of minimal non-trivial blocks.) A minimal block system is one that is not the refinement of any block system other than the trivial partition of Ω into one set Ω.

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315

The group G is primitive if it has no block systems with more than 1 and fewer than |Ω| blocks, i.e., no block systems other than (a) the partition of Ω into the single set Ω and (b) the partition of Ω into one-element sets. It follows from the definitions that a group G ≤ Sym(Ω) acts as a primitive group on any minimal block system. 2.7

Tools from the theory of permutation groups

The following result guarantees the existence of an element of small support in a group under rather mild conditions. It is essentially due to Wielandt. Thanks are due to L. Pyber for the reference. Lemma 2.15 For any  > 0, there are C1 , C2 ≥ 0 such that the following holds. Let n ≥ C1 . Let G  Sym(Ω), |Ω| = n, be a group containing a section isomorphic to Alt(k) for k ≥ C2 log n. Then there is a g ∈ G, g = e, such that | supp(g)| < n. This Lemma replaces [22, Lem. 3.19], which was based on [7, Lem. 3]. Proof Suppose there is no g ∈ G, g = e, such that | supp(g)| < n; we say G has minimal degree at least n. We can assume without loss of generality that C2 log n ≤ k ≤ 2C2 log n, since, given that G contains a section isomorphic to Alt(k), it contains a section isomorphic to Alt(k  ) for all k  ≤ k. Let ω = min(, 0.4). Then, by [14, Thm. 5.5A], we have n > ks for s = μ(k + 1) , μ = (1 − ω)1/5 . By Stirling’s formula, k    1 k 1 k! (ω √ = s!(k − s)! s k μμ (1 − μ)(1−μ) for k greater than a constant depending only on μ. This gives a contradiction with C2 log n ≤ k ≤ 2C2 log n for n ≥ C1 when C1 and C2 are large enough √ in terms of . (We use the condition k ≤ 2C2 log n to ensure that the effect of 1/ k is negligible.)  We also need a result telling us that a large subset of Sym(Σ) generates a large symmetric subgroup in a few steps. Lemma 2.16 Let H  Sym(Σ), |Σ| = k. Let ρ ∈ (1/2, 1). If |H| ≥ ρk k! and k is larger than a constant depending only on ρ, then there exists an orbit Δ ⊂ Σ of H such that |Δ| ≥ ρ|Σ| and H|Δ is Alt(Δ) or Sym(Δ). Proof By [14, Thm. 5.2B], which is a somewhat strengthened version of [24, Lem. 1.1]. We simply need to check that      1 k k [Sym(Σ) : H] < min , 2 k/2 m for m = )ρk*. This inequality follows from Stirling’s formula for k larger than a constant depending on ρ. 

316 2.8

Helfgott: Growth in linear algebraic groups and permutation groups Diameter comparisons: directed and undirected graphs

We wish to derive a (version of) Theorem 1.3, which is a bound on the diameter of a directed graph, from Theorem 1.4, which is a statement on sets A satisfying A = A−1 . It would be natural to expect such a statement to imply only a bound on the diameter of an undirected graph. As it happens, the distinction between directed and undirected graph matters little in this context, thanks to the following result. Lemma 2.17 ([3], Thm. 1.4) Let G be a finite group and A a set of generators of G. Then

2 diam Γ(G, A) ' (log |G|)3 · diam Γ G, A ∪ A−1 , where the implied constant is absolute. It is thus enough to prove Theorem 1.3 (and analogous statements) for sets S satisfying S = S −1 : simply replace S by S ∪ S −1 , and use Lemma 2.17.

3

Finding few generators for a transitive group

Let us be given a set A ⊂ Sym(Ω). When is it the case that there is a small subset C A of (A ∪ A−1 ∪ {e})n (say) that generates A , or at least a transitive subgroup of A ? Can we put further conditions on the elements of A , such as, for instance, that they all be conjugates of each other? Our motivation for considering this question is the following. We will find it necessary to do what amounts to bounding the size of the intersection of a slowly growing set A with the centralizer of an element of large support. It would stand to reason that there should be stronger bounds than for the intersection of A with a subgroup without long orbits: being in the centralizer is a more restrictive condition. Given one of our main tools (Lemma 2.3; see in particular the remark between its proof and its statement), we can quickly reduce this task to the following one: given an element h of large support and a set of generators A of G, find g1 , . . . , gk ∈

 A ∪ A−1 ∪ {e} such that C(h) ∩ g1 C(h)g1−1 ∩ . . . ∩ gk C(h)gk−1

(3.1)

is equal to {e}, or at least very small. It is clearly enough for the group h, g1 hg1−1 , . . . , gk hgk−1

(3.2)

to be transitive: the centralizer H of a transitive subgroup of Sym(n) is semiregular (that is, no element of H other than e fixes any point), and thus has ≤ n elements.

 Let us, then, show that there are g1 , . . . , gk ∈ A ∪ A−1 ∪ {e} , k and small, such that the group (3.2) is transitive. We will be able to prove what we want with k ' log log n, assuming that A is 4-transitive.

Helfgott: Growth in linear algebraic groups and permutation groups

317

(As we will later discuss, reaching the bound k ' log n is substantially easier. An analogous, but not identical, result with k ∼ log n can be found in [5, Lemma 5.13].) Our proof will be by iteration, with the iterative step being given by the next proposition. We will be working with partitions into orbits, but will prove the proposition for general partitions. Recall that, given two partitions P , Q of a set Ω, the join P ∨ Q is the finest partition that is coarser (not necessarily strictly so) than both P and Q. The trivial partition of Ω is the partition {Ω}. Given a partition P of Ω and an element x ∈ Ω, we define SP (x) to be the element of P containing x. Let sP (x) = |SP (x)|. The total variation distance between two probability measures μ1 , μ2 on a finite set X is defined to be δ(μ1 , μ2 ) = max |μ1 (S) − μ2 (S)|. S⊂X

Thus, for instance, μ is at total variation distance at most  from the uniform distribution if μ(S) = |S|/|X| + O∗ () for every S ⊂ X. Suppose this is the case. Then, given any function f : X → R with 0 ≤ f (x) ≤ T for every x ∈ X, we can easily estimate the expected value Eμ f (x) of f (x) with respect to μ: clearly, >

T

f (x) =

1L(f,t) (x)dt

(“layer-cake decomposition”),

0

where L(f, t) = {x ∈ X : f (x) ≥ t}, and so >

T

Eμ (f (x)) =

>

T

Probμ (x ∈ L(f, t))dt =

0

0



 L(f, t) dt + O∗ () dt. |X|

Applying the same idea to the uniform distribution, without the error term O∗ (), we obtain that 1  Eμ (f (x)) = f (x) + O∗ (T ). (3.3) |X| x∈X

Lemma 3.1 Let P be a partition of a finite set Ω with |Ω| = n. Let m ≥ 2. Denote by ρ the proportion of elements x of Ω such that sP (x) ≥ m. Let g ∈ Sym(Ω) be taken at random with a distribution such that, for any element v of the set Ω(2) of ordered pairs of distinct elements of Ω, the probability distribution of v g is at total variation distance ≤  from the uniform distribution −1 on Ω(2) . Assume that the same is true for the probability distribution of v g as well. 1. With positive probability, the proportion of elements x in Ω such that sP ∨P g (x) ≥ m is ≥ 1 − (1 − ρ)2 − . 2. Assume that  ≤ ρ/100, n ≥ 100 and 2 ≤ m ≤ n/2. Then, with positive probability, the proportion of elements x of Ω such that sP ∨P g (x) ≥ (1+ρ/3)m is ≥ ρ2 /8.

318

Helfgott: Growth in linear algebraic groups and permutation groups

3. Assume that  ≤ min(ρ/25, ρ/4m) and n ≥ 250. Then, with positive probability, P ∨ P g contains at least one set of size at least   ρ− 2 ρ n, m . min 10 2 The proof is straightforward, in that we will proceed by taking expected values. We are giving constants simply for concreteness; they have not been optimized. Conclusion (2) is substantially weaker than what we could obtain by means of more complicated variance-based arguments such as those we will use in the proof of Lemma 3.2. Proof Let B be the set of all x ∈ Ω such that sP (x) < m. For each x ∈ B, the probability that xg ∈ B is ≤ |B|/n +  = 1 − ρ + . Hence, the expected value of the number of x ∈ B such that xg ∈ B is ≤ (1 +  − ρ)|B| = (1 +  − ρ)(1 − ρ)n ≤ ((1 − ρ)2 + )n. It obviously follows that the number of such x is ≤ ((1 − ρ)2 + )n with positive probability. In other words, conclusion (1) holds. For each x ∈ Ω such that sP (x) ≥ m, we choose a subset Z(x) ⊂ SP (x) of size m, in such a way that, for every set S in P , every element of S is contained in exactly m sets Z(x), x ∈ S. (For instance, we may identify each element of P having m ≥ m elements with the set Z/m Z, and then let Z(x) = {x, x + 1, . . . , x + m − 1} mod m for every x ∈ Z/m Z. We can easily see that every element of Z/m Z is then contained in exactly m sets Z(x).) For every x ∈ Ω such that sP (x) < m, we let Z(x) = ∅. We write z(x) for |Z(x)|; we see that z(x) can take only the values m or 0. We see immediately that  9 9   9Z(x) ∩ Z(x )9 = |{x ∈ SP (x) : y ∈ Z(x )}| x,x ∈Ω

x∈Ω y∈Z(x)

=

 

m = ρm2 n,

(3.4)

x∈Ω y∈Z(x)

a fact that will be useful:later.;g −1 We write Zg (x) for Z xg , and zg (x) for |Zg (x)|. By definition, : : −1 ;;g = SP g (x). Zg (x) ⊂ SP xg Clearly, sP ∨P g (x) ≥ z(x) + zg (x) − |Z(x) ∩ Zg (x)| . For any x,

(3.5)

: −1 :9 : −1 ;9; ; 9 9 E(zg (x)) = E 9Z xg 9 = m · Prob xg ∈ B = m · (ρ + O∗ ()),

since B is the set of elements x of Ω such that Z(x) = ∅, and |Z(x)| = m for all x ∈ Ω \ B.

Helfgott: Growth in linear algebraic groups and permutation groups

319

Given x ∈ Ω \ B and a y ∈ Z(x), we should estimate the probability that y is an element of Z(x) ∩ Zg (x), where g is, as always, taken at random. Evidently, if y = x, then y ∈ / Zg (x) if Zg (x) is empty, and y ∈ Zg (x) otherwise. If y = x, we have y ∈ Zg (x) if and only if g −1 sends (x, y) to an element of S = {(x , y  ) : x ∈ Ω \ B, y  ∈ Z(x), y  = x }. The number of elements of S is |Ω \ B| · (m − 1). Hence Prob(y ∈ Zg (x)) ≤

ρm |Ω \ B| · (m − 1) +≤ + . n(n − 1) n

Therefore, E (|Z(x) ∩ Zg (x)|) ≤ Prob(x ∈ Zg (x)) +



Prob(y ∈ Zg (x))

y∈Z(x) y =x

≤ρ++

  ;  : ρm m2 + ≤ρ 1+ + m, n n

y∈Z(x) y =x

and so

⎛

⎞  1 E⎝ (zg (x) − |Z(x) ∩ Zg (x)|)⎠ ρn x∈Ω\B      1 m2 m · (ρ − ) − ρ 1 + ≥ − m |Ω \ B| n x∈Ω\B   m2 − 2m. = ρm − ρ 1 + n

(3.6)

Thus, with positive probability,   m2 1  − 2m. (zg (x) − |Z(x) ∩ Zg (x)|) ≥ ρm − ρ 1 + ρn n x∈Ω\B

The contribution of all x ∈ Ω \ B such that zg (x) − |Z(x) ∩ Zg (x)| ≤ ρm/3 is at most ρm/3. Each one of the other x ∈ Ω \ B contributes at most m/ρn. Hence, the number of all x ∈ Ω \ B such that zg (x) − |Z(x) ∩ Zg (x)| > ρm/3 is : ;; : 2 m2 ρ − 2m m − 1 + 3 n ≥ m/ρn      1 m 1 2 ρ2 2 2 − + − ≥ ρ n − 2ρn ≥ ρ2 n − 2ρn ≥ n, 3 m n 6 n 8 where we use the assumptions 2 ≤ m ≤ n/2,  ≤ ρ/100, n ≥ 100. By (3.5), we obtain that conclusion (2) holds. It remains to prove conclusion (3).

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Helfgott: Growth in linear algebraic groups and permutation groups

Let x ∈ Ω. For each y ∈ SP (x), every element of Zg (y) lies in SP g (y) and hence in SP ∨P g (x). Therefore, by inclusion-exclusion, for U ⊂ SP (x) arbitrary, 9 9 9 9   9 9 9 9 9Zg (y) ∩ Zg (y  )9 . 9 SP g (y)99 ≥ zg (y) − (3.7) |SP ∨P g (x)| ≥ 9 9y∈U 9 y∈U y,y  ∈U y =y 

By our assumptions on the distribution of g, ⎛ ⎞    : : −1 ;; E⎝ zg (y)⎠ = E (zg (y)) = E z yg y∈U

y∈U

≥



y∈U

(3.8)

(ρ − )m = (ρ − )m|U |,

y∈U

and, similarly,

⎛ E⎝



⎞ zg (y)⎠ ≤ (ρ + )m|U |.

(3.9)

y∈U

We can apply (3.3) to X = Ω(2) and f (x, x ) = |Z(x)∩Z(x )|, with the probability −1 distribution on X given by (y, y  )g , where (y, y  ) is a given element of Ω(2) and g is taken randomly in the sense we have been using throughout. Then, by (3.4) and the fact that 0 ≤ f (x, x ) ≤ m for all (x, x ) ∈ X, 9 ρm2 n − mn 9 + O∗ (m). E 9Zg (y) ∩ Zg (y  )9 = n(n − 1) Hence ⎛

⎞

  ⎜  9 9⎟ ρm2 n − mn ⎜ 9Zg (y) ∩ Zg (y  )9⎟ + O∗ (m) . E⎜ ⎟ = |U |(|U | − 1) n(n − 1) ⎠ ⎝  y,y ∈U y =y 

(3.10)  ≤

 ρm2 + m · |U |2 . n

Therefore, by (3.7),  E (sP ∨P g (x)) ≥ (ρ − )m|U | −

 ρm2 + m |U |2 . n

In general, the maximum of an expression at − bt2 , a, b > 0, is of course attained when t equals t0 = a/2b; moreover, since a(t0 − Δ) − b(t0 − Δ)2 = a2 /4b − bΔ2 , we see that at0 − b(t0 )2 ≥ a2 /4b − b. We let a = (ρ − )m, b = ρm2 /n + m.

Helfgott: Growth in linear algebraic groups and permutation groups

321

Suppose first that t0 > m. Then we simply choose any x ∈ Ω with |SP (x)| ≥ m, and choose U ⊂ SP (x) with |U | = m. Since t0 = a/2b > m, we see that am − bm2 = am(1 − bm/a) > am/2, and so E (sP ∨P g (x)) >

ρ− 2 am = m . 2 2

Now suppose that t0 ≤ m. Then there is an x ∈ Ω such that |SP (x)| ≥ t0 . We choose U ⊂ SP (x) with |U | = t0 , and obtain that

a2 E sP ∨P g (x) ≥ − b. 4b Clearly 1/(r1 + r2 ) ≥ min(1/2r1 , 1/2r2 ) for a, b > 0. Hence   n a2 1 min , ρm2 8 ρm2 m n + m  

ρ2 (1 − /ρ)2 (1 − /ρ)2 min ρn, m ≥ min ρn, 4ρm2 , ≥ 8  8

a2 = 4b

a2 /4

≥

where we use the assumption  ≤ ρ/4m. Again by  ≤ ρ/4m, we see that t0 = a/2b ≤ m implies that (1 − )ρm = a ≤ 2bm = 2

ρm3 ρm3 ρm + 2m2 ≤ 2 + , n n 2

and so 4ρm2 ≥ (1 − 2)ρn. Therefore   ρm2 a2 (1 − /ρ)2 −b≥ (1 − 2)ρn − + m 4b 8 n (1 − /ρ)2 1 − 2 ≥ (1 − 2)ρn − ρ − m. 8 4 If m ≥ ρn/10, then P contained sets of size ≥ ρn/10 to begin with, and hence so does P ∨ P g . If m < ρn/10, we obtain that  

a2 (1 − /ρ)2  1 − 2 E sP ∨P g (x) ≥ −b≥ (1 − 2) − − ρn 4b 8 10 4n   1 1 ρn (24/25)2 23 · − − n ≥ 0.10031ρn > ≥ 8 25 250 600 10 by the assumptions  ≤ ρ/25 ≤ 1/25 and n ≥ 150. Thus, conclusion (3) holds.  Simply using Lemma 3.1 repeatedly, we could give a proof of Prop. 3.3 with k in the order of log n. Our crucial induction step, allowing us k ' log log n, will be provided by the following Lemma. The proof will proceed by variance-based bounds. (In other words, we will be using Chebyshev’s inequality.)

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Helfgott: Growth in linear algebraic groups and permutation groups

Lemma 3.2 Let P be a partition of a finite set Ω with |Ω| = n. Let m ≥ 2. Denote by ρ the proportion of elements x of Ω such that sP (x) ≥ m. Let g ∈ Sym(Ω) be taken at random with a distribution as in Lemma 3.1, with  ≤ min(1/1000, ρm/n). √ Assume that ρ ≥ 999/1000 and 1000 ≤ m ≤ n/100. Then, with positive probability, m2 sP ∨P g (x) ≥ 2 for more than n/2 elements x of Ω. We will use several estimates in the proof of part (3) of Lemma 3.1. We shall use the same notation as in that proof: Z(x), z(x), Zg (x) and zg (x) are the same as there. Proof  Let fg (x) = y∈Z(x) zg (y). By (3.8) and (3.9), (ρ − )m · z(x) ≤ E(fg (x)) ≤ (ρ + )m · z(x). Therefore, writing EF =

1 n



x∈Ω F (x),

we see that

(ρ − )ρm2 ≤ E Efg ≤ (ρ + )ρm2 ,

(3.11)

where we take g at random, as always. Let Rg =

1 n



9 9 9Zg (y) ∩ Zg (y  )9 .

(3.12)

x∈Ω y,y  ∈Z(x) y =y 

Then, by (3.10),  E (Rg ) ≤ ρ

 ρm2 ρ2 m4 2ρ2 m4 + m m2 = + ρm3 ≤ , n n n

(3.13)

where we use the assumption  ≤ ρm/n.  Let us now bound the expected value of x∈Ω fg (x)2 . Clearly fg (x)2 =



zg (y)zg (y  )

y,y  ∈Z(x)

=



zg (y)2 +



zg (y)zg (y  ).

y,y  ∈Z(x) y =y 

y∈Z(x)

Now, for x such that Z(x) is non-empty, ⎛ ⎞   E⎝ zg (y)2 ⎠ ≤ (ρ + )m2 = (ρ + )m3 y∈Z(x)

(3.14)

y∈Z(x)

(3.15)

Helfgott: Growth in linear algebraic groups and permutation groups ⎛

and

323

⎞

⎜  ⎜ E⎜ ⎝ 

y,y ∈Z(x) y =y 

⎟ ⎟ zg (y)zg (y  )⎟ ≤ ⎠



(ρ2 + )m2

y,y  ∈Z(x)

(3.16)

y =y  2

= (ρ + )m2 · (m2 − m). Therefore, ? E

1 fg (x)2 n

@ ≤ (ρ + )ρm3 + (ρ2 + )ρ(m4 − m3 )

x∈Ω

(3.17)

= (ρ − ρ )ρm + (ρ + )ρm . 2

3

2

4

We have just established a bound on the expectation of the variance: for ? @2 1 1 2 Vf = f (x) − f (x) n n x∈Ω

(3.18)

x∈Ω

we quickly see, by (3.11) and (3.17), that

E Vf g

@2 ⎞  1 =E fg (x) ⎠ − E⎝ n x∈Ω x∈Ω

2 2 3 2 4 ≤ (ρ − ρ )ρm + (ρ + )ρm − (ρ + )ρm2 ?

1 fg (x)2 n

⎛?

@

(3.19)

≤ (1 − ρ)ρ3 m4 + 1 m4 , where

(1 − ρ)ρ2 . (3.20) m We may call Vfg the variance of fg , just as we may call Efg the expectation of fg . Now we should give a bound on the variance V Efg of Efg . Clearly ⎛? ⎛ ⎞ @2 ⎞ : ;   1 1 E Ef2g = E ⎝ fg (x) ⎠ = 2 · E ⎝ fg (x)fg (x )⎠ . (3.21) n n  1 = ρ +

x,x ∈Ω

x∈Ω

By the definition of fg (x) and the fact that, for every y ∈ Ω with Z(y) = ∅, y ∈ Z(x) for exactly m values of x ∈ Ω,      fg (x)fg (x ) = zg (y)2 + zg (y)zg y  x,x ∈Ω

x,x ∈Ω y∈Z(x)∩Z(x )

=

 y∈Ω Z(y) =∅

m2 zg (y)2 +



x,x ∈Ω y∈Z(x),y  ∈Z(x ) y =y 

m2 zg (y)zg (y  ).

y,y  ∈Ω Z(y),Z(y  ) =∅ y =y 

(3.22)

324

Helfgott: Growth in linear algebraic groups and permutation groups

Much as in (3.15) and (3.16), ⎛

⎞

⎜  ⎟  ⎜ ⎟ E⎜ m2 zg (y)2 ⎟ ≤ m2 · (ρ + )m2 = (ρ + )ρm4 n ⎝ ⎠ y∈Ω Z(y) =∅

and

y∈Ω Z(y) =∅



m2 zg (y)zg (y  ) ≤

y,y  ∈Ω Z(y),Z(y  ) =∅ y =y 



m2 · (ρ2 + )m2

y,y  ∈Ω Z(y),Z(y  ) =∅ y =y  2 2 4 2

≤ (ρ + )ρ m n . Hence

; (ρ + )ρ : m4 + (ρ2 + )ρ2 m4 , E Ef2g ≤ n

and so, by (3.11),

(ρ + )ρ 4 V Ef g ≤ m + ((ρ2 + ) − (ρ − )2 )ρ2 m4 ≤ 2 m4 , n

(3.23)

where (ρ + )ρ . n ; : :

2

2 ; . Here, of course, V Efg = E Ef2g − E Efg = E Efg − E Efg 2 = (1 + 2ρ)ρ2  +

(3.24)

By (3.11), (3.13), (3.19) and (3.23) and Cauchy-Schwarz, we conclude that ; :

2 E Ef2g − c1 Vfg − c2 Efg − E Efg − c 3 n · Rg

2

≥ E Efg − c1 E Vfg − c2 V Efg − c3 n · E (Rg ) ≥ Km4 for any c1 , c2 , c3 > 0, where

K = (ρ − )2 ρ2 − (1 − ρ)ρ3 + 1 c1 − 2 c2 − 2ρ2 c3 . We will choose c1 , c2 , c3 so that K is positive. Then the probability that V fg ≤

Ef2g c1

E fg ≥

,

√

9

9 c 2 9E fg − E E f g 9 ,

nRg ≤

Ef2g c3

(3.25)

will be positive. What happens when (3.25) is the case? √ 1. First of all, Efg ≥ c2 |Efg − E(Efg )| implies √

√

√

c2 c2 E Ef g ≤ Ef g ≤ √ E Efg . c2 + 1 c2 − 1

(3.26)

Helfgott: Growth in linear algebraic groups and permutation groups

325

2. By Chebyshev’s inequality, if (3.25) is the case, then for any τ > 0, the number of x ∈ Ω such that 1−τ ≤

fg (x) ≤1+τ Ef g

does not hold is at most (nVfg /Ef2g )/τ 2 ≤ n/c1 τ 2 . 3. By (3.12) and the last inequality in (3.25), for any τ  > 0, the number of x ∈ Ω such that  9 9 9Zg (y) ∩ Zg (y  )9 ≤ τ  Ef g y,y  ∈Z(x) y =y 

does not hold is ≤ Rg n/τ  Efg ≤ Ef2g /c3 τ  Efg = Efg /c3 τ  . Hence, for ≥ (1 − 1/c1 τ 2 )n − Efg /c3 τ  values of x ∈ Ω, by (3.7) and (3.11), √

c2 |SP ∨P g (x)| ≥ fg (x) − τ  Efg ≥ (1 − τ − τ  )Efg ≥ √ (1 − τ − τ  )E Efg c2 + 1 √ c2 (1 − τ − τ  )(ρ − )ρm2 . ≥√ c2 + 1 (3.27) Moreover, by (3.11) and (3.26), √ √ Ef g c2 E(Efg ) c2 (ρ + )ρ 2 ≤ ≤ m . √ √ c3 τ  c2 − 1 c3 τ  c2 − 1 c3 τ  √ Thus, by the assumption m ≤ n/100, we obtain that   Ef g 1 1− n− ≥ ρ n c1 τ 2 c3 τ  √ c2 (ρ + )ρ 1 ρ =1− −√ . c1 τ 2 c2 − 1 1002 c3 τ 

for



(3.28)

It is time to choose the parameters c1 , c2 , c3 , τ and τ  . We let c1 =

1 , 4δ1

δ1 =

(1 − ρ)ρ3 + 1 , ρ4

c2 =

1 , 42

c3 =

ρ2 , 8

(3.29)

and τ = 1/4, τ  = 1/12. Then K ≥ (ρ − )2 − 3/4 > 0, by our assumptions on ρ and . In fact, since we are assuming  ≤ 1/1000, m ≥ 1000, ρ ≥ 999/1000 and n ≥ (100m)2 ≥ 109 , 1−ρ ≤ 0.001001, 1 ≤  + m 2 ≤ 3 +

(1 + ) ≤ 0.0030001, n

326

Helfgott: Growth in linear algebraic groups and permutation groups δ1 ≤ ρ−1 − 1 +

1 1000 0.001001 ≤ ≤ 0.002007 −1+ ρ4 999 (999/1000)4

by (3.20), (3.24), and (3.29). Hence, by (3.27), √ c2 (1 − τ − τ  )(ρ − )ρm2 sP ∨P g (x) = |SP ∨P g (x)| ≥ √ c2 + 1   1 1 1 998 999 2 m √ 1− − m > ≥ 4 12 1000 1000 2 1 + 42 for at least ρ n elements x of Ω. Moreover, by (3.28), ρ ≥ 1 −

1+ 4δ1 1 √ − τ2 1 − 42 1002 · ρ2 · 8

1 12

1 > 0.86 > . 2 

Proposition 3.3 Let P be a partition of a finite set Ω with |Ω| = n. Assume that at least ≥ ρn elements of Ω, ρ > 0, lie in sets in P of size > 1. Let A ⊂ Sym(Ω) be a set of generators of a 4-transitive subgroup of Sym(n). Let h ∈ Sym(Ω) have support of size n − c, where 0 ≤ c < n.

v Then there are g1 , . . . , gk ∈ A ∪ A−1 ∪ {e} , k = O(log log n) + Oρ,c (1), v = O(n10 ), such that the partition Qk defined by gj hg −1

Qj = Qj−1 ∨ Qj−1 j

Q0 = P,

for 1 ≤ j ≤ k

is the trivial partition of Ω. An example given by W. Sawin4 suggests that 4-transitivity is a necessary assumption.

v Proof Let A0 ⊂ A ∪ A−1 be as in Lemma 2.8, and let g ∈ Av0 ⊂ A ∪ A−1 be the outcome of a random walk of length v, where v = )n9 log(n4 /)* for a given  > 0. Then, by Prop. 2.7 applied to the Schreier graph Γ(G, A0 ; Ω(4) ), given any two elements v1 , v2 of the set Ω(4) of quadruples of 9distinct elements of 9 9 Ω,9 the probability that g takes v1 to v2 lies between (1 − )/ 9Ω(4) 9 and (1 + )/ 9Ω(4) 9. A moment’s thought shows that, since the support of h is of size n − c, the number of quadruples (r, s, r , s ) ∈ Ω(4) such that rh = r and sh = s is at least (n − c)(n − c − 3) and at most (n − c)(n − c − 1). Given any (x, y, x , y  ) ∈ Ω(4) , the probability that ghg −1 takes (x, y) to (x , y  ) equals the probability that g takes (x, y, x , y  ) to a tuple (r, s, r , s ) such that rh = r and sh = s , and so lies between (1 − )(n − c)(n − c − 3)/|Ω(4) | and (1 + )(n − c)(n − c − 1)/|Ω(4) |. Therefore, for any (x, y) ∈ Ω(2) and any subset S ⊂ Ω(2) , the probability that −1 (x, y)ghg ∈ S is at least (1 − ) 4

(n − c)(n − c − 3) 9 9 (|S| − 4n). 9Ω(4) 9

On MathOverflow, in comments to [21].

(3.30)

Helfgott: Growth in linear algebraic groups and permutation groups

327

(Here |S| − 4n is a lower bound for the number of pairs in S not containing x or y.) We bound (3.30) from below by (1 − ) where

(n − c)(n − c − 3) |S| 4n(n − c)(n − c − 3) |S| − ≥ 99 (2) 99 −  , (n − 2)(n − 3) n(n − 1) n(n − 1)(n − 2)(n − 3) Ω  4(n − c)(n − c − 3) (n − c)(n − c − 3) +  =+ 1− (n − 2)(n − 3) (n − 1)(n − 2)(n − 3) c c 4n 8 + 3c ≤+ + + ≤+ , n − 3 n (n − 1)(n − 2) n 



(3.31)

by n ≥ 6. Since we can apply the same bound to the complement of S, we conclude −1 that the distribution of (x, y)ghg is at total variation distance at most  from the uniform distribution. We can apply the same statement to h−1 instead of h. Hence, we can apply Lemmas 3.1 and 3.2 with ghg −1 instead of g, and  instead of , provided that their conditions on  , ρ, m and n hold. If n is bounded by a constant C, say, then we can proceed as follows: at every step, we look at an element L of Qj of maximal length (size), and let g be as above, with  = 1/100, say. (We could even take v = )n5 log(n2 /)*, and look only at the Schreier graph Γ(G, A; Ω).) Then any given element of L is sent to any given element of Ω \ L with positive probability, and so, trivially, L becomes larger in Qj+1 = Qj ∨ Qgj with positive probability, i.e., for at least one g. We set gj+1 equal to that g. After at most k = C steps, we obtain, then, that Qk consists of a single set, equal to Ω, and so we are done. We can assume, then, that n is greater than a constant C. We start by applying parts (1) and (2) of Lemma 3.1 a bounded number of times so that we get to a √ state in which either the conditions of Lemma 3.2 are fulfilled or m > n/50. (If the conditions are already fulfilled, then, of course, this initial stage may be skipped.) The initial value m0 of m will be 2. We let ρ0 = ρ. We let  = ρ/4000; since we can assume that n ≥ 4000(8 + 3c)/ρ, we obtain from (3.31) that  ≤ ρ/4000 + (8 + 3c)/n ≤ ρ/2000. In particular, the condition  ≤ ρ/100 in part (2) of Lemma 3.1 is satisfied. 1. We begin by applying part (1) of Lemma 3.1 repeatedly, with m held constant. At the jth step, Lemma 3.1 guarantees us the existence of a gj ∈ Av0 such −1 that, for Qj = Qj−1 ∨ Qghg j−1 , the proportion ρj of elements x ∈ Ω for which sQj (x) ≥ m satisfies ρj−1 + ρ2j−1 2000 : ρj ; = (1 − ρj ) 1 − 2

1 − ρj ≤ (1 − ρj−1 )2 +  ≤ 1 − 2ρj−1 + ≤1−

3ρj−1 ρ2j−1 + 2 2

provided that ρj−1 ≤ 999/1000. Thus, letting j be at least about log 1000 9

9 9log 1 − ρ0 9 2

328

Helfgott: Growth in linear algebraic groups and permutation groups (which is O(1/ρ0 )), we obtain a new value ρj of ρ such that ρj ≥ 999/1000, say.

2. If m ≥ 1000, we stop. Otherwise, we apply part (2) of Lemma 3.1. We are guaranteed the existence of an element gj+1 ∈ Av0 such that, for Qj+1 = −1 Qj ∨Qghg , the proportion of elements x ∈ Ω such that sQj (x) ≥ (1+ρj /3)m j is at least ρ2j /8. We choose that g, let our new values mj+1 and ρj+1 of m and ρ be   A: ρ2j ρj ; B 333 (999/1000)2 1 mj+1 = 1 + mj ≥ 1 + mj , ρj+1 = ≥ > , 3 1000 8 8 9 and go back to step 1. It is clear that, after s = O(1/ρ) + O(log max(c, 1)) steps, we obtain a partition Qs such that the proportion of elements x of Ω satisfying sQs (x) ≥ 1000 · max(c, 1) is at least 999/1000. Now – and here we are at the heart of the proof of this proposition – we go again through an iterative procedure, only we will now be alternating a bounded number of applications of part (1) of Lemma 3.1 and an application of Lemma 3.2, rather than O(1/ρ) applications of part (1) of Lemma 3.1 and an application of part (2) of Lemma 3.1. Throughout the iteration, ρ stays bounded from below by 1/2. We let  = 1/n, so that, by (3.31),  ≤  +

9 + 3c 8 + 3c = , n n

(3.32)

and thus  ≤ 1/1000 and  ≤ 500 max(c, 1)/n ≤ ρm/n both hold. The conditions √ of Lemma 3.2 are thus satisfied for as long as m ≤ n/100. The important part this time is that Lemma 3.2 enables us to take mj+1 = )m2j /2*, rather than mj+1 = )(1 + ρ/3)mj * as in Lemma 3.1(2). Thanks to this fact, after only O(log log n) steps, we obtain a partition Qs such that the proportion √ of elements x of Ω satisfying sQs (x) > n/100 is at least 999/1000. √ We are almost done. We apply part (3) of Lemma 3.1 with m = ) n/100*, ρ = 999/1000 and  = 1/n (and thus  as in (3.32). We obtain a partition Qs +1 = Qs ∨ Qgs containing at least one set of size ≥ min(ρn/9, (ρ − )m2 /2) > ((998/1000)/20000)n > n/20100. Tautologically, the proportion ρ of elements of Ω lying in that set is > 1/20100. We then alternate a bounded number of applications of part (1) of Lemma 3.1 and an application of part (2) of the same Lemma, iterating a bounded number of times, to obtain a partition Qs such that at least n/2 of its elements lie in sets of size > n/2. Finally, we apply part (1) (with  = 1/n) O(log log n) times, and obtain a partition Qs such that the proportion x of elements of Ω lying in sets in the partition of size > n/2 is at least 1 − 2 ≥ 1 − (18 + 6c)/n. Since |Ω| = n, there can be at most one set in the partition of length > n/2; that is, at least n − (18 + 6c) elements of Ω lie in one and the same set S. We now proceed as in the case of n bounded, choosing at each step a g ∈ Al0 that increases the size of the orbit S by at least 1. After a bounded number of

Helfgott: Growth in linear algebraic groups and permutation groups

329

steps, we obtain that all elements of Ω lie in the same set of the partition, i.e., the final partition consists of the single set Ω.  We come to the main result of this section. Proposition 3.4 Let Ω be a finite set of size |Ω| = n. Let g0 ∈ Sym(Ω) have support of size ≥ αn, α > 0. Let A ⊂ Sym(Ω) with A 4-transitive. 6 Then there are γi ∈ (A ∪ A−1 ∪ {e})n , 1 ≤ i ≤ , where = O((log n)/α), and −1 v gi ∈ (A ∪ A ∪ {e}) , 1 ≤ i ≤ k, v = O(n10 ), k = O(log log n), such that, for h = γ1 g0 γ1−1 · γ2 g0 γ2−1 · · · γ g0 γ−1 ,

(3.33)

h, g1 hg1−1 , g2 hg2−1 , . . . , gk hgk−1

(3.34)

the group

is transitive. Proof Let γ1 , . . . , γ be as in Lemma 2.13 (applied with g0 instead of g and A ∪ A−1 ∪ {e} instead of A), so that the element h defined in (3.33) has support of size n − c with c = 0 or c = 1. (If n is less than a constant, we do not need apply Lemma 2.13; we can simply let h = g0 , as c = n − | supp(h)| will be bounded.) Write h as a product of disjoint cycles, and let P be the partition of Ω given by the cycles. We can now apply Proposition 3.3 with ρ = (n − 1)/n and c = 0 or c = 1. It is clear, inductively, that, for 0 ≤ j ≤ k, Qj is finer (not necessarily strictly) than the partition of P given by the orbits of h, g1 hg1−1 , g2 hg2−1 , . . . , gj hgj−1 . Since Qk is the trivial partition, it follows that the group in (3.34) has a single orbit. 

4

Babai-Seress revisited

The proof of the main result in [8] has what looks like a bookkeeping mistake, or rather two mistakes, at the very end ([8, p. 242], “Proof of Theorem 1.4”): the right side of the last displayed equation has a factor of diam(Alt(m(G))) where it should have a product of squares of several such factors. We will show how to fix the result and its proof. (The fact that [8] could not be right at this point was first pointed out to the author by L. Pyber; he also shared ideas that can be used in addressing the gap, including one followed and mentioned below.) The intermediate result [8, Thm. 2.3], is in fact correct. We will prove it again here (§4.1), in part for the sake of clarity, and in part so as to give an improved version (Prop. 4.7). We will be following [8, §3–4] quite closely.

330 4.1

Helfgott: Growth in linear algebraic groups and permutation groups Imprimitivity and structure trees

A tree is a graph without cycles, with one vertex labeled as the root vertex, or root for short. The vertices at level j are those at distance j from the root. The leaves of a tree are the vertices at maximal distance from the root. That maximal distance is called the height h of the tree. A child of a vertex v at level j, 0 ≤ j < h, is a vertex at level j + 1 connected to v by an edge. A descendant of v is a child of v, or a child of a child, etc. The following definition has its origin in the study of algorithms on permutation groups, and in particular [27]. It provides a convenient way to work with permutation groups that may not be primitive. Definition 4.1 Let G ≤ Sym(Ω) be a transitive permutation group. A structure tree T for (G, Ω) is defined as follows. The set of leaves is Ω. If G is primitive, then it consists of a root vertex and, for each leaf, an edge connecting the root to the leaf. If G is not primitive, we choose a maximal block system B1 , . . . , Bk , define a structure tree for G as a group acting transitively on B1 , . . . , Bk , and then draw edges from the vertex corresponding to each Bi to the elements of Bi (which then become the leaves). It is clear that G has a natural action on T . Given a vertex v of T , we define the stabilizer Gv ≤ G to be the setwise stabilizer of the block corresponding to v (or the stabilizer of the element of Ω corresponding to v, if v is a leaf). If w is a descendant of v, then Gw is a subgroup of Gv . For v a vertex that is not a leaf, define Kv to be the intersection ∩w Gw , where w ranges over all children of v. It is clear that Kv is a normal subgroup of Gv . It is also clear that Gv /Kv acts primitively on the set of children of v, due to the maximality of the block systems used in Definition 4.1. It is easy to see from the definition that G acts transitively on all vertices of T at  a given level. It follows that the normal core Nj = g∈G gGv g −1 of the stabilizer Gv of a vertex v depends only on the level of v. Indeed, it is the intersection ∩w Gw , where w ranges over all w at the same level as v. Moreover, if the level j of a vertex  v is less than the height of the tree (i.e., v is not a leaf), then the normal core g∈G gKv g −1 of Kv equals Nj+1 . Part of the reason for working with the groups Gv , rather than just with Nj , is that Gv acts transitively on the block corresponding to v, whereas Nj may not. The following lemmas are quoted as “folklore” in [8, §3]. Lemma 4.2 Let H be a subgroup of a direct product of simple groups M1 × M2 × . . . × Mk such that the projection πi : H →  Mi is surjective for every 1 ≤ i ≤ k. Then H is isomorphic to a direct product i∈I Mi , where I ⊂ {1, 2, . . . , k}. Proof If the projection φ : H → M2 × M3 × . . . × Mk is injective, we apply the lemma to φ(H) ⊂ M2 × . . . Mk and are done, by induction. Suppose φ is not injective. Let h1 , h2 ∈ H be distinct elements such that φ(h1 ) = φ(h2 ). Then −1 φ(h−1 1 h2 ) = e, and so h = h1 h2 lies in H ∩ M1 . Let g ∈ M1 be arbitrary.  There is an element g of H mapped to g by π1 ; conjugating h by it, we obtain g  h(g  )−1 = ghg −1 , which must then lie in H ∩ M1 . Since we can do as much for

Helfgott: Growth in linear algebraic groups and permutation groups

331

every g ∈ M1 , we conclude that H ∩M1 must contain the subgroup of M1 generated by all elements of the form ghg −1 . Since M1 is simple, that subgroup is precisely M1 , and so H ∩ M1 = M1 . Let K = H ∩({e}×M2 ×. . . Mk ). Since H ∩M1 = M1 , we know that H ∼ M1 ×K. For each 2 ≤ i ≤ k, the image πi (K) is invariant under conjugation by πi (H) = Mi , and thus must be either {e} or Mi . We eliminate all indices i for which πi (K) = {e}, and apply the Lemma inductively to K as a subgroup of the direct product of the remaining Mi .  Lemma 4.3 Let H1  H2 ≤ G, H2 /H1 simple. Let Ni = ∩g∈G gHi g −1 . Then N2 /N1 is isomorphic to a direct product of copies of H2 /H1 . Proof We may assume that N2 = N1 . By the second isomorphism theorem, H1 ∩ N2 is a normal subgroup of H2 ∩ N2 = N2 , and N2 /(H1 ∩ N2 ) is isomorphic to N2 H1 /H1 , which is a normal subgroup of H2 /H1 . Since H2 /H1 is simple, that subgroup is either trivial or all of H2 /H1 . If it were trivial, then N2 ≤ H1 , and so N2 ≤ gH1 g −1 for every g ∈ G; it would follow immediately that N2 = N1 ∩N2 = N1 . Thus, we may assume that N2 /(H1 ∩ N2 ) is isomorphic to H2 /H1 . The same argument applied to any conjugate gH1 g −1 , g ∈ G, instead of H shows that N2 /(gH1 g −1 ∩ N2 ) is isomorphic to H2 /H1 . Now, N2 /N1 is isomorphic to its image under the natural map N2 /N1 → (N2 /(gH1 g −1 ∩ N2 ))g∈G , since N1 =  −1 ∩ N ). We apply Lemma 4.2, and obtain that N /N is isomorphic 2 2 1 g∈G (gH1 g to a direct product of groups of the form N2 /(gH1 g −1 ∩ N2 ) ∼ H2 /H1 .  The following is an extremely useful (and by now standard) consequence of the Classification Theorem and the O’Nan-Scott theorem. This is the one way in which the Classification Theorem is needed for the proof of our results. Proposition 4.4 ([13], [25]) Let G ≤ Sym(Ω) be a primitive group, where |Ω| = n. Then either 1. |G| ≤ nO(log n) , or 2. there is a subgroup N  G, [G : N ] ≤ n, isomorphic to a direct product

r Alt(m)r = Alt(m) × . . . × Alt(m), where r ≥ 1, m ≥ 5 and n = m for k some 1 ≤ k ≤ m − 1. Proof Just a few words on how the statement follows from [25, Main Thm.]. Case (ii) there asserts that there is a set Δ ⊂ Ω with |Δ| < 9 log2 n such that G(Δ) = {e}; since [G : G(Δ) ] ≤ n|Δ| , it follows immediately that conclusion (1) holds. Assume, then, that we are in case (i) in [25, Main Thm.]; that case gives us r a subgroup m

r N as in (2) here. It also gives us thatr m ≥ 2, [G : N ] ≤ 2 r! and n= k for some 1 ≤ k ≤ m − 1. Clearly, n ≥ m . If m ≥ max(2r, 5), then [G : N ] ≤ (2r)r ≤ mr ≤ n, and so we obtain conclusion (2). If m < 2r, then, since r ≤ logm n ≤ log2 n, we see that |N | = (m!/2)r ≤ mmr < nm < nmax(2r,5) ≤ nO(log n) , whereas [G : N ] ≤ 2r r! ≤ (2r)r ≤ nO(log n) . Hence |G| = nO(log n) , that is, conclusion (1) holds. 

332

Helfgott: Growth in linear algebraic groups and permutation groups

The motivation for wanting subgroups Alt(m) with m ≥ 5 is of course that Alt(m) is then simple. We could do without the following bound5 , in that using a trivial bound in case (2) of Prop. 4.4 would be enough for our purposes; our intermediate results would become somewhat weaker, but our final results (Thms. 1.4 and 6.1) would not be affected. At the same time, there is no reason to avoid the lemma we are about to state. It does use the Classification Theorem, but only in the sense that it uses [25, Main Thm.] (or rather the version with sharp constants in [28]). Lemma 4.5 ([16, Thm. 1.3]) Let G ≤ Sym(Ω) be a primitive group, where |Ω| = n. Let {e} = H0  H1  . . .  H = G. Then

≤

8 log n 4 − . 3 log 2 3

The trivial bound assuming Prop. 4.4, would be ' (log n)2 . Lemma 4.6 ([2]) The length of any subgroup chain {e} = H0  H1  . . .  H = Sym(n) of Sym(n) is at most 2n − 3. The trivial bound would be ≤ (log | Alt(n)|)/ log 2 = O(n log n). We will now prove a version of [8, Thm. 2.3]. Proposition 4.7 Let G ≤ Sym(Ω) be transitive, |Ω| = n. Then G has a series of normal subgroups {e} = H0  H1  . . .  H = G, Hi  G, with {1, . . . , } being partitioned into two sets, A, B, such that these properties hold: 1. each quotient Hi+1 /Hi is a direct product of at most 2n copies of a simple group Mi , 2. for each i ∈ A, the group Mi is an alternating group Alt(mi ) with mi ≥ 5,  3. m = i∈A mi satisfies m ≤ n, 4. if G = Alt(Ω) and G = Sym(Ω), then mi ≤ n/2 for every i ∈ A,  5. i∈B |Mi | = (n/m)O(log(n/m)) · m = nO(log n) , 6. = O (log n). All implied constants are absolute. The series {e} = H0  H1  . . .  H = G is a refinement of the series {e} = N1  . . .  Nh = G defined by the structure tree as above. Proof We construct a structure tree as in Def. 4.1. We choose a leaf v and denote by v0 , v1 , . . . , vh−1 , vh = v all vertices on the path from the root v0 to v. For 0 ≤ i ≤ h − 1, let Gi = Gvi /Kvi . Apply Prop. 4.4 with Gi instead of G, and with the setof children of vi instead of Ω. Write ni for the number of children of vi . Clearly, 0≤i≤h−1 ni = n, and so h ≤ (log n)/ log 2 = O(log n). 5

Thanks are due to D. Holt for the reference.

Helfgott: Growth in linear algebraic groups and permutation groups

333

If conclusion (1) holds, we simply take a composition series {e} = Si,0  Si,1  . . .  Si,i = Gi of Gi . By Lemma 4.5, i ' log ni . Write S˜i,j for the preimage of Si,j under the map Gvi → Gi = Gvi /Kvi . Let g S˜i,j g −1 . Hi,j = g∈G

By Lemma 4.3, for 1 ≤ j ≤ i , Hi,j /Hi,j−1 is a direct product of copies of the simple group Mi,j = S˜i,j /S˜i,j−1 = Si,j /Si,j−1 . By Lemma 4.6, there are at most 2n such copies. Evidently,  O(log ni ) |Mi,j | = |Gi | = ni . 1≤j≤i

We include every one of the groups Mi,j in the set B (to be implicitly redefined as a set of indices at the end of the proof). If conclusion (2) of Prop. 4.4 holds, we write ri for r, mi for m and ki for k, and let {e} = Si,ri  Si,ri +1  . . .  Si,i = Gi /N be a composition series of Gi /N . Since |Gi /N | ≤ ni , we see that i = ri + O(log ni ) = O(log ni ). Given that N ∼ Alt(mi )ri , we can write {e} = Ai,0  Ai,1  . . .  Ai,ri = N, where Ai,j /Ai,j−1 ∼ Alt(mi ) for 1 ≤ j ≤ ri . This time, we define A˜i,j to be the preimage of Ai,j under the map Gvi → Gi = Gvi /Kvi , and S˜i,j to be the image of Si,j under the composition Gvi → Gi → Gi /N . Let  g A˜i,j g −1 if 0 ≤ j ≤ ri , Hi,j = g∈G ˜ −1 if ri < j ≤ i . g∈G g Si,j g We let Mi,j = Si,j /Si,j−1 for ri < j ≤ i , and Mj,1 = Ai,j /Ai,j−1 ∼ Alt(mi ) for 1 ≤ j ≤ ri . i ri We know from Prop. 4.4 that m ≤ ni , where 1 ≤ ki ≤ mi − 1. If ri ≥ 2, then ki √ mi ≤  ni ≤ ni /2 ≤ n/2; if ri = 1 and ki ≥ 2, then mi (mi − 1) ≤ 2ni and so, since mi ≥ 5, mi ≤ ni /2 ≤ n/2. If ri = 1 and ki = 1, then mi = ni . In that case, if G is not primitive, then mi = ni ≤ n/2, whereas, if G is primitive, mi = ni = n and so Alt(n) ≤ G. For all 1 ≤ j ≤ i , Mi,j is simple. By Lemma 4.3, for 1 ≤ j ≤ i , Hi,j /Hi,j−1 is a direct product of copies of the simple group Mi,j ; by Lemma 4.6, there are at most 2n such copies. We include Mi,j ∼ Alt(mi ) in A for 1 ≤ j ≤ ri and Mi,j in B for ri < j ≤ i . It is clear – whether conclusion (1) or (2) holds – that, for any i less than the height h of our tree, Hi,0 = ∩g∈G gKvi g −1 = Ni+1 , whereas Hi,i = ∩g∈G gGvi g −1 = Ni . Hence we can define our subgroups H0 , H1 , . . . , H to be Hh−1,0 , Hh−1,1 , . . . , Hh−1,h−1 = Hh−2,0 , . . . , Hh−2,h−2 , . . . H1,1 = H0,0 , . . . , H0,0 .

334

Helfgott: Growth in linear algebraic groups and permutation groups

The (trivial) accounting is left to the reader.  4.2

Reduction of the diameter problem to the case of alternating groups

First, a lemma essentially due to Schreier. The statement is as in [8, Lemma 5.1]. Lemma 4.8 (Schreier) Let G be a finite group. Let N  G. Then diam G ≤ (2 diam(G/N ) + 1) diam(N ) + diam(G/N ) ≤ 4 diam(G/N ) diam(N ). Proof Let us be given a set A = {g1 , . . . , gr } of generators of G. Write d1 for diam(G/N ), d2 for diam(N ) and m for |G/N |. Then, by the definition of diameter, there are σ1 , . . . , σm ∈ (A ∪ A−1 ∪ {e})d1 giving us a full set of representatives of G/N . As is well-known, S = {σi gj σk−1 : 1 ≤ i, k ≤ m, 1 ≤ j ≤ r} ∩ N is a set of generators of N (Schreier generators). Hence, N = (S ∪ S −1 ∪ {e})d2 , and so G = {σ1 , . . . , σm } · N ⊂ (A ∪ A−1 ∪ {e})d1 · (S ∪ S −1 ∪ {e})d2 ⊂ (A ∪ A−1 ∪ {e})d1 +(2d1 +1)d2 .  Corollary 4.9 Let G be a finite group. Let {e}  H1  H2  . . .  H = G. Then diam(G) ≤ 4−2

−1 

diam (Hi+1 /Hi ) .

i=0

Proof Immediate from Lemma 4.8.



Lemma 4.10 ([8, Lemma 5.4]) Let G = T1 × T2 × · · · × Tn , where the Ti are nonabelian simple groups. Let diam(Ti ) = di , d = maxi di . Then diam(G) ' n3 d2 . We will go over the ideas of the proof of the Lemma in a moment, when we improve it in the special case of the alternating group (Lemma 4.14). L. Pyber pointed out to the author that the dependence on d could and should be improved so as to be linear; the quadratic dependence of Lemma 4.10 on d is one of the gaps in the proof of the main result in [8]. It is actually enough to improve the dependence on d in the alternating case. The tool we will use is a simple lemma, similar to [8, Prop. 5.8]. Lemma 4.11 Let g ∈ Alt(Ω), g = e, |Ω| ≥ 4. Then there is an h ∈ Alt(Ω) such that [g, h] is either a 3-cycle or a product of two disjoint 2-cycles.

Helfgott: Growth in linear algebraic groups and permutation groups

335

Here [g, h] denotes the commutator g −1 h−1 gh. Proof Write g as a product of disjoint cycles. If g contains two disjoint 3-cycles (abc), (def ), let h equal (adc)(bef ). Then [g, h] = (af )(bd). If g contains two disjoint 2-cycles (ab)(cd), let h be the 3-cycle (abc); their commutator [g, h] will be (ac)(bd). If g contains a k-cycle (abcd . . . ), k ≥ 4, let h = (abc). Then [g, h] = (adc). Finally, if g consists of a single 3-cycle (abc), let d be an element of Ω different from a, b and c, and define h to be (bcd). Then [g, h] = (ad)(bc).  The following lemma is of course extremely familiar. Lemma 4.12 Let n ≥ 5. Then every element of Alt(Ω), |Ω| = n, can be written as (a) the product of at most n − 1 3-cycles, (b) the product of at most (n + 1)/2 elements of the form (ab)(cd). Proof We prove part (a) by induction. If g ∈ Alt(Ω) is not the identity, then there is an a ∈ Ω such that b = ag is distinct from a, and another c = a, b that  is also in the support of g. Then, for g  = g · (bac), we see that ag = a and   supp(g ) ⊂ supp(g), and so | supp(g )| ≤ | supp(g)| − 1. We prove part (b) in the same way: if | supp(g)| ≥ 4, there are distinct a, b, c, d such that ag = b and cg = d; then, for g  = g · (ba)(dc), | supp(g  )| ≤ | supp(g)| − 2. If | supp(g)| = 3, then g is a 3-cycle (abc), and so, for b , c not in the support of g, g equals the product of (ac)(b c ) and (bc)(b c ).  The following result is classical, easy and very well-known. According to [23], it was first proved in [29]. Lemma 4.13 Let m ≥ 5. Then every element of Alt(m) is a commutator, i.e., expressible in the form [x, y], x, y ∈ Alt(m). Now we come to the proof of an improved version of Lemma 4.10 in the case of the alternating group. Lemma 4.14 Let G = T1 × T2 × · · · × Tn , where the Ti are alternating groups Alt(mi ), mi ≥ 5. Let diam(Ti ) = di , d = maxi di , m = maxi mi . Then diam(G) ' n3 md. L. Pyber suggests using [26] to prove an analogous improvement on Lemma 4.10 for arbitrary finite simple groups Ti . Proof Let S be a set of generators of G, and let A = S ∪ S −1 ∪ {e}. Write πi : G → Ti for the projection of G to Ti . By the definition of d, πi Ad = Ti for every 1 ≤ i ≤ m. The set A2d+1 ∩ ker(πi ) must then contain a set of generators of ker(πi ) (namely, Schreier generators). In particular, for any j = i, A2d+1 ∩ ker(πi ) contains at least one element gi,j such

that πj (gi,j ) = e. By Lemma 4.11 and πj Ad = Tj , there is an h ∈ Ad such that πj ([g, h]) ∈ πj (A6d+2 ) is either a 3-cycle or the product of two disjoint 2-cycles. Hence, conjugating [g, h] by all elements of Ad , we obtain either all 3-cycles in Tj

336

Helfgott: Growth in linear algebraic groups and permutation groups

or all products of disjoint 2-cycles in Tj . By Lemma 4.12, we can express every element of Tj as a product of (a) at most mj − 1 3-cycles, (b) at most (mj + 1)/2 3-cycles. At the same time, [g, h] is in ker(πi ), and so, obviously, are its conjugates. Hence, for Bi = A(8d+2)mj ∩ ker(πi ), we see that πj (Bi ) = Tj . Now, say that, for S, S  ⊂ {1, . . . , n} \ {j}, there are sets BS , BS  ⊂ Ak satisfying πj (BS ) = πj (BS  ) = Tj as well as BS ⊂ ker(πi ) for every i ∈ S and BS  ⊂ ker(πi ) for every i ∈ S  . Then BS∪S  = {[x, y] : x ∈ BS , y ∈ BS  } is a subset of A4k contained in ker(πi ) for every i ∈ S ∪ S  . Moreover, by Lemma 4.13, πj (BS∪S  ) = Tj . We apply this procedure repeatedly, first expressing Zj = {1, . . . , n} \ {j} as the union of two disjoint sets S, S  of size (n − 1)/2 and )(n − 1)/2*, respectively, and then doing a recursion, expressing at each point the set we are given as the union of two disjoint sets of sizes differing by at most 1, until we reach the single log n 2 element sets S = {i}, i = j. We obtain a subset BZj of A4 2 k ⊂ A4n k , where k = (8d + 2)mj , such that BZj ⊂ ker(πi ) for every i = j, and πj (BZj ) = Tj . (Here we note that 4log2 n ≤ 4log2 n+1 ≤ 4 · 4log2 n = 4n2 .) 3 Multiplying the sets BZj , we obtain that A4n (8d+2)m contains all of G.  We will also need a very easy analogue for abelian simple groups. Lemma 4.15 Let G = Z/pZ × Z/pZ × · · · × Z/pZ (n times). Then diam(G) ≤ np/2 . Proof Let S be a set of generators of G, and let A = S ∪ S −1 ∪ {e}. We can see G both as a group and as a vector space (which we may call V ) over Z/pZ. We choose a non-identity element v of A. Trivially, every element of the linear span v of v can be written in the group as v j for some j ∈ Z with |j| ≤ (p − 1)/2. We project the elements of A to A mod v , and thus reduce the problem to that for the space V modv instead of V = G.  We finally come to the fixed (and improved) version of [8, Thm. 1.4]. Proposition 4.16 Let  G ≤ Sym(Ω), |Ω| = n, be transitive. Then there are m1 , . . . , mk ≥ 5 with ki=1 mi ≤ n such that diam(G) ≤ nO(log n)

k 

diam(Alt(mi )).

i=1

Moreover, 1. for every 1 ≤ i ≤ k, G has a composition factor isomorphic to Alt(mi ), 2. if G = Alt(Ω), Sym(Ω), then mi ≤ n/2 for every 1 ≤ i ≤ k. Proof Apply Proposition 4.7. By Cor. 4.9, diam(G) ≤ 4O(log n)

−1  i=0

diam(Hi+1 /Hi ).

Helfgott: Growth in linear algebraic groups and permutation groups By Lemma 4.10, Lemmas 4.14–4.15 and Prop. 4.7(1),  n3 mi diam(Mi ) diam(Hi+1 /Hi ) ' n3 diam(Mi )2 Hence diam(G) ≤ 4O(log n) (O(n3 ))



337

if i ∈ A, if i ∈ B.

(mi diam(Mi ))



diam(Mi )2 .

i∈B

i∈A

Trivially, diam(Mi ) ≤ |Mi |, and so, by Prop. 4.7(5)   diam(Mi ) ≤ |Mi | ≤ nO(log n) , i∈B

whereas, by Prop. 4.7(2), 

i∈B

diam(Mi ) =

i∈A

Finally, by Prop. 4.7(3) and (6),

5





diam(Alt(mi )).

i∈A i∈A mi

≤ n and ' log n.



Main argument

Let us set out to prove our main result (Theorem 1.4). Part of the general strategy will be as in [22], but much simplified. Throughout, A ⊂ Sym(Ω), |Ω| = n, with A = A−1 , e ∈ A, and A is 3transitive. We assume that log |A| ≥ C(log n)3 , where C > 0 is a constant large enough for our later uses. 5.1

Existence of a large prefix

For j = 1, 2, . . . , we choose distinct elements α1 , α2 , . . . ∈ Ω such that, for j = 1, 2, . . . , 9 9 9 (A4 )(α1 ,α2 ,...,αj−1 ) 9 9α 9 ≥ ρn, (5.1) 9 j 9 where we set ρ = e−1/5 = 0.818 . . . , say. We stop when (A4 )(α1 ,α2 ,...,αk ) has no orbits of size ≥ ρn. Inequality (5.1) holds for 1 ≤ j ≤ k. Let Σ = {α1 , . . . , αk−1 }. By Corollary 2.12 (applied with Σ ∪ αk instead of Σ), either (2.8) holds, and we are done, or k≥

log |A| . 15(log n)2

(5.2)

We can assume henceforth that (5.2)

holds. By Lemma 2.14, the restriction A8(k−1) Σ |Σ is a subset of Sym(Σ) with at least

ρk−1 (k − 1)! elements. Let A = A8(k−1) Σ , H = A . If, as we may assume, k is larger than an absolute constant, then, by Lemma 2.16, there exists an orbit Δ ⊂ Ω of H|Σ , such that |Δ| ≥ ρ · (k − 1) and H|Δ contains Alt(Δ). Thus, in particular, H has a section isomorphic to Alt(k − 1), namely, the quotient defined by restricting either H or a subgroup of H of index 2 to Δ.

338 5.2

Helfgott: Growth in linear algebraic groups and permutation groups The case of descent

Applying Lemma 2.15 with  = 1/8, we see that H contains an element g0 =  e such that | supp(g0 )| < n/8, assuming, as we may, that n is greater than an absolute constant and that k − 1 ≥ C2 log n, where C2 is an absolute constant. (A4 )

Let O = αk (Σ) . We know that (5.1) holds for j = k, i.e., |O| ≥ ρn. Denote by  O the orbit of H containing O. Suppose first that either |O | ≤ e−1/10 n or H|O = Alt(O ), Sym(O ). Define D to be the diameter of H|O . If the diameter of Γ(H, A ) is no larger than D, then g0 ∈ (A )D ⊂ A8kD . If, on the other hand, diam Γ(H, A ) > D, then there is an element g of H that is in (A )D+1 but not in (A )D . At the same time, since D is the diameter of H|O , there is an element h of (A )D whose restriction h|O equals g|O . Clearly, g −1 h is non-trivial, lies in (A )2D+1 and has trivial restriction to O . By this last fact, the support of g −1 h is of size ≤ (1 − ρ)n < n/5. Therefore, in either case, there exists an element g  of (A )2D+1 ⊂ A8k(2D+1) with support < n/5. (Many thanks are due to Henry Bradford for spotting a gap at this point in a previous version of this paper, and for the alternative argument we have just given.) By Lemma 2.9, diam(Γ(A , A ∪ g  )) ' n8 (log n)O(1) , and so diam(Γ(A , A)) ' 8k(2D + 1)n8 (log n)O(1) ' n10 D. Thus we attain conclusion (1.4) in Theorem 1.4. We call this case the case of descent. 5.3

The case of growth

Assume henceforth that H|O is either Alt(O ) or Sym(O ) and that |O | ≥ e−1/10 n. We can then of course assume that |O | ≥ 6, and so the action of H on O is 4transitive. Let B = (A2 )(α1 ,α2 ,...,αk ) . Then, by (5.1), every orbit of BB −1 is of length < ρn = e−1/5 n ≤ e−1/10 |O |. We apply Cor. 2.11 with A |O instead of A, B|O instead of B, O instead of Ω, |O | instead of n and e−1/10 instead of ρ, and obtain that there is a g ∈ (A )m ⊂ A8km , m ' n6 log n, such that 1/10 9 2 2 −1 9 9B gB g 9 ≥ |B|1+ log n .

Since g is in the setwise stabilizer of Σ, we know that B 2 gB 2 g −1 is a subset of 16km+8 A . Therefore, by Lemma 2.4, (Σ) 9 9 9: 9 99 A16km+8 99 1/10 9 32km+16 ; 9 1 (Σ) 1+ 9 A 9≥ ≥ |B| log n . 9 9 n n (α1 ,...,αk )

Helfgott: Growth in linear algebraic groups and permutation groups

339

We have obtained what we wanted: growth in a subgroup – namely, the subgroup Sym(Ω)(α1 ,...,αk ) of Sym(Ω). We apply Lemma 2.5 with Sym(Ω)(α1 ,...,αk ) instead of H and 32km + 16 instead of k, and obtain that 1/10 9 9 9 32km+17 9 |B| log n |A|. 9≥ 9A n

(5.3)

Only one thing remains: to ensure that |B| is not negligible compared to |A|. By Lemma 2.5, 9 9 |A| 9 9 |B| = 9 A2 (α1 ,...,α ) 9 ≥ k . k n C Thus, if k ≤ (logn |A|)/2, we see that |B| ≥ |A|, and so 9 9 1 1 9 32km+17 9 1 1+ 20 log 1+ 21 log n ≥ |A| n, 9A 9 ≥ |A| n say, yielding a strong version of conclusion (1.3). Assume from now on that k > (logn |A|)/2. Since (5.1) holds for j = k, and since we can assume that ρn > 1, there is at least one non-trivial element of (A4 )(Σ) . Call it g0 . If supp(g0 ) ≤ n/4, then, by Lemma 2.9, diam(Γ(Sym(Ω), A)) ≤ 4 diam(Γ(Sym(Ω), (A4 )(Σ) ∪ A)) ' n8 (log n)c , and we are done. Assume, then, that supp(g0 ) > n/4, and so | supp(g0 ) ∩ O | > n/4 − (1 − e−1/10 )n ≥ n/7 ≥ |O |/7. Now we can finish the argument in any of two closely related ways. One way would involve combining Prop. 3.4 with Lemma 2.3, as we said at the beginning of §3. However, we will find it simpler to proceed in a way closer to the procedure explained in [22, §1.5]. We apply Prop. 3.4 with O instead of Ω and (A )|O instead 6 of A. We obtain γi ∈ (A ∪ (A )−1 ∪ {e})n , 1 ≤ i ≤ , where = O(7 log n) = O(log n), and g1 , . . . , gk ∈ (A )v , v = O(n10 ), k  = O(log log n), such that, for ; : 6 h = γ1 g0 γ1−1 · γ2 g0 γ2−1 · · · γ g0 γ−1 ∈ A4+2·8(k−1)n , (Σ)

the group

h, g1 hg1−1 , g2 hg2−1 , . . . , gk hgk−1 

acts transitively on O . Write h0 = h, hi = gi hgi−1 for 1 ≤ i ≤ k  . Since h fixes Σ pointwise and gi fixes Σ;setwise, hi fixes Σ pointwise for every 0 ≤ i ≤ k  . Thus, : 6 +2v 4+2·8(k−1)n hi ∈ A . By the same argument, the map (Σ)

φ : g → (gh0 g −1 , gh1 g −1 , . . . , ghk g −1 ) sends (A )2 ⊂ (A16k )Σ to a subset of the Cartesian product ; ; : : 6 6 A32k+4+2·8(k−1)n +2v × . . . × A32k+4+2·8(k−1)n +2v (Σ)

(k  + 1 times). (Σ)

340

Helfgott: Growth in linear algebraic groups and permutation groups

Moreover, two elements g, g  satisfy φ(g) = φ(g  ) if and only if g −1 g  hi (g −1 g  )−1 for every 0 ≤ i ≤ k  , i.e., if and only if g −1 g  lies in C(h0 , h1 , . . . , hk ). We know that h0 , h1 , . . . , hk acts transitively on O . It is easy to show that an element of the centralizer of a transitive group can have a fixed point if and only if it is the identity. Thus, if two distinct g, g  ∈ ((A )2 )αk satisfy φ(g) = φ(g  ), then, since g(g  )−1 fixes αk , it must act as the identity on O . In other words, g(g  )−1 is

 −1/10 a non-identity element of support of size ≤ n − |O | ≤ 1 − e n ≤ n/10. We can now apply Lemma 2.9 (Babai-Beals-Seress), and obtain that diam(Γ(A , A)) ' 8kn8 (log n)4 ' n10 . ((A )2 )αk is injective. Then 9k +1 9 9 9 9 ≥ 9((A )2 )αk 9 (Σ) 9

Assume, then, that the restriction of φ to 9: 9 32k+4+2·8(k−1)n6 +2v ; 9 A 9 and so

   1 9 9 9 9 1 |A | k +1 9 N 9 9  2 k +1 9 ≥ 9 A (Σ) 9 ≥ ((A ) )αk n 10 6 for N = 32k + 4 + 2 · 8(k − 1)n + 2v = O n . Since Σ = {α1 , . . . , αk−1 } and |A | ≥ ρk−1 (k − 1)!, where ρ = e−1/5 , it follows that 9 9 9 1 9 99 AN 9

(|A |/n) k +1 (Σ) 9 9 9 2N ≥ ≥ 9 A (α1 ,...,αk ) 9 n n

1 k−1 1 ρ (k − 1)!/n k +1 (ρk k!) k +1 ≥ . ≥ n n2 Since B = (A2 )(α1 ,...,αk ) , we know from Lemma 2.5 that 9 9 9 2N 9 9 2N +1 9 9 A (α1 ,...,αk ) 9 9≥ 9A · |A|. |B| Hence either

1

(ρk k!) 2k +2 |A| n  or |B| ≥ (ρk k!)1/(2k +2) /n. In the latter case, by (5.3), |A2N +1 | ≥

9 9 9 32km+17 9 9A 9≥

?

1

(ρk k!) 2k +2 n

1 @ 10 log n

(5.4)

1 |A| |A| ( (ρk k!) 20(k +1) log n . n n

(5.5)

The amount on the right in (5.4) is clearly greater than that on the right in (5.5), so we can focus on bounding the right side of (5.5) from below. By ρ = e−1/5 , Stirling’s formula, and the assumptions that log |A| ≥ C(log n)3 (or even just log |A| > C(log n)2 ) and k > (logn |A|)/2,  ρk k! (

k e6/5

k

 ≥

log |A| 2e6/5 log n

 log |A| 2 log n

log |A|

≥ (log |A|) 4 log n = |A|

log log |A| 4 log n

.

Helfgott: Growth in linear algebraic groups and permutation groups

341

Hence, again by log |A| ≥ C(log n)3 , log log |A|

1

log log |A| (ρk k!) 10k log n |A| O((log n)2 log log n) ≥ ≥ |A| O((log n)2 log log n) . n n Taking N  = max(2N + 1, 32km + 17) = O n10 , we conclude that

9 9 log log |A| 1+ 9 N9 9A 9 ≥ |A| O((log n)2 log log n) . Theorem 1.4 is thus proved.

6

Iteration

We can now prove a marginally weaker version of Theorem 1.3. The reader will notice that the proof we are about to give works for any 3-transitive group G, not just for G = Alt(n) and G = Sym(n). However, by [31, Cor. to Thm. A], every 3-transitive and in fact every 2-transitive group G on n elements that is not Alt(n) or Sym(n) has exp(O((log n)3 )) elements. Thus, in such a case, the result we are about to prove would be trivial. Theorem 6.1 Let G = Alt(n) or Sym(n). Let S be a set of generators of G. Then diam Γ(G, S) ≤ eK(log n)

4 (log log n)2

,

(6.1)

where K is an absolute constant. Since | Alt(n)| ≥ n!/2 ( (n/e)n , it follows immediately that, for G = Alt(n) and for G = Sym(n), diam Γ(G, S) ≤ eO((log log |G|)

4 (log log log |G|)2 )

,

where the implied constant is absolute. Proof We can assume that e ∈ S. By Lemma 2.17, we can assume that S = S −1 as well.  For any k ≥ 1, if Sk = S k+1 ,9 then S k = S k for every k  > k, and so S k = S = 9 9 9 G. So, if S k = G, 9S k+1 9 ≥ 9S k 9 +1. Applying this statement for k = 1, 2 . . . , m, we see that, for any m, |S m | ≥ min(m, |G|). Let A0 = S m for m = )exp C(log n)3 *, where C is as in the statement

of Thm. 1.4. Then, assuming n is larger than a constant, |A0 | ≥ exp C(log n)3 . (If n is not larger than a constant, then the theorem we are trying to prove is trivial.) We apply Theorem 1.4 to A0 instead of A. If conclusion (1.4) holds, we stop. If C conclusion (1.3) holds, we let A1 = An0 and apply Theorem 1.4 to A1 . We keep on iterating until conclusion 1.4 holds, and then we stop. We thus have A0 , A1 , . . . , Ak , C k ≥ 0, such that Ai+1 = Ani for 0 ≤ i ≤ k − 1, |Ai+1 | ≥ |Ai |

1+c

log log |Ai | (log n)2 log log n

(6.2)

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(i.e., conclusion (1.3) holds) for 0 ≤ i ≤ k − 1, and conclusion (1.4) holds for Ak . Let us bound k. Write ri = log |Ai |. By (6.2),   log ri ri+1 = 1 + c ri . (log n)2 log log n We also know that r0 ≥ 2 (or really rather more) and rk ≤ log |G|. The number of steps needed for ri to double is E D (log n)2 log log n 2(log n)2 log log n ≤ ≤ , c log ri c log ri where we use the fact that )y* ≤ 2y for y ≥ 1 and we assume, as we may, that c ≤ 1. We conclude that k is at most (2/c)(log n)2 log log n times  r=2j r≤log |G|

1 = log r

 0≤j≤log2 log |G|

1 j log 2

' log log log |G| ' log log n. Write this bound in the form k ≤ C  (log n)2 (log log n)2 .

Ck We see that Ak ⊂ An0 = Al0 = S lm for l ≤ exp CC  (log n)3 (log log n)2 and, as before, m ≤ exp(C(log n)3 ). (The author would like to thank L. Pyber profusely for pointing out that, as we have just seen, the presence of log log |A| in the exponent in (1.3) means we save a factor of (log n)/ log log n in the bound on k.) By conclusion (1.4), which holds for Ak , diam(Γ(G, S)) ≤ lm · diam(Γ(G, Ak )) ≤ lmnC diam(G ), where G is a transitive group on n ≤ n elements such that either (a) n ≤ e−1/10 n or (b) G  Alt(n ), Sym(n ). If n ≤ e−1/10 n and either G ∼ Alt(n ) or G ∼ Sym(n ), then diam(G ) ≤ max(diam(Sym(n )), diam(Alt(n ))) ≤ 4 diam(Alt(n ))

(6.3)

by Lemma 4.8. If G  Alt(n ), Sym(n ), then, we apply Prop. 4.16, and obtain that diam(G) ≤ (n )C



log n

k  i=1

diam(Alt(mi )) ≤ eC

 (log n)2

k 

diam(Alt(mi )),

(6.4)

i=1

 where ki=1 mi ≤ n ≤ n, mi ≤ n /2 ≤ n/2 for every 1 ≤ i ≤ k, and C  is   2  3 an absolute constant. Clearly, lmnC max(4, eC (log n ) ) ≤ eC (log n) for C  an 3/2 absolute constant, provided that (say) n ≥ e .

Helfgott: Growth in linear algebraic groups and permutation groups

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We can assume, as an inductive hypothesis, that Theorem 6.1 is true for G1 = Alt(n1 ), n1 ≤ e−1/10 n. In other words, diam(G1 ) ≤ eK(log n1 )

4 (log log n

1)

2

.

If (6.3) above applies, we let n1 = n , and obtain that 

3

 4

2

 2

diam(Γ(G, S)) ≤ eC (log n) (log log n) eK(log n ) (log log n )  3 4 2 ≤ e(C (log n) +K((log n)−1/10) )(log log n) . For K > (10/3.99) · C  (say) and n larger than a constant,   1 4 C  (log n)3 + K (log n) − ≤ K(log n)4 , 10 and so Theorem 6.1 is true for n. If (6.4) applies instead, then diam(Γ(G, S)) ≤ eC

 (log n)3 (log log n)2

k 

eK(log mi )

4 (log log m

i)

2

,

i=1

 where ki=1 mi ≤ n and mi ≤ n/2 for all 1 ≤ i ≤ k. If k = 1, we proceed as above, with 1/2 instead of e−1/10 . If k > 1, then, assuming, as we may, that m1 ≥ mi for all 2 ≤ i ≤ k, k 

(log mi )4 = (log m1 )4 +

i=1

k 

(log mi )4 ≤ (log m1 )4 + (

i=2

= (log m1 )4 + (log

k 

log mi )k

i=2 k 

mi )4 ≤ (log m1 )4 + (log n − log m1 )4

i=2

≤ (log 2)4 + (log n/2)4 , since 2 ≤ m1 ≤ n/2. Hence, much as above, diam(Γ(G, S)) ≤ eC

 (log n)3 (log log n)2

eK((log n−log 2)

4 +(log 2)4 )(log log n)2

.

For K > 1/(3.99 log 2) and n larger than a constant, C  (log n)3 + K(((log n) − log 2)4 + (log 2)4 ) ≤ K(log n)4 , and so Theorem 6.1 is true for n in this case as well.



References [1] L. Babai, On the order of doubly transitive permutation groups. Invent. math. 65(3) 1981/82, 473–484.

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[2] L. Babai, On the length of subgroup chains in the symmetric group. Comm. Algebra 14(9) 1986, 1729–1736. [3] L. Babai, On the diameter of Eulerian orientations of graphs. In: Proc. 17th ACMSIAM Symp. on Discrete Algorithms. ACM, New York, 2006, 822–831 ´ Seress, On the diameter of the symmetric group: polynomial [4] L. Babai, R. Beals, and A. bounds. In: Proc. 15th ACM-SIAM Symp. on Discrete Algorithms. ACM, New York, 2004, 1108–1112. ´ Seress, Fast management of permutation groups. I. [5] L. Babai, E. M. Luks, and A. SIAM J. Comput. 26(5) 1997, 1310–1342. [6] L. Babai, N. Nikolov, and L. Pyber, Product growth and mixing in finite groups. In: Proc. 19th ACM-SIAM Symp. on Discrete Algorithms, ACM, New York, 2008, 248–257. ´ Seress, On the degree of transitivity of permutation groups: a short [7] L. Babai and A. proof. J. Combin. Theory Ser. A 45(2) 1987, 310–315. ´ Seress, On the diameter of permutation groups. Eur. J. Comb. [8] L. Babai and A. 13(4) 1992, 231–243. [9] J. Bourgain and A. Gamburd, Uniform expansion bounds for Cayley graphs of SL2 (Fp ). Ann. of Math. (2) 167(2) 2008, 625–642. [10] J. Bourgain, A. Gamburd, and P. Sarnak, Affine linear sieve, expanders, and sumproduct. Invent. math. 179(3), 2010, 559–644. 3 [11] J. Bourgain, A. Gamburd, and P. Sarnak, Generalization of Selberg’s 16 theorem and affine sieve. Acta Math. 207(2) 2011, 255–290. [12] E. Breuillard, B. Green, and T. Tao. Approximate subgroups of linear groups. Geom. Funct. Anal. 21(4), 2011, 774–819. [13] P. J. Cameron, Finite permutation groups and finite simple groups. Bull. London Math. Soc. 13(1), 1981, 1–22. [14] J. D. Dixon and B. Mortimer. Permutation groups, Volume 163 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1996. [15] A. Eskin, Sh. Mozes, and H. Oh, On uniform exponential growth for linear groups. Invent. math. 160(1) 2005, 1–30. [16] S. P. Glasby, C. E. Praeger, K. Rosa, and Gabriel Verret, Bounding the composition length of primitive permutation groups and completely reducible linear groups. Preprint. Available as https://arxiv.org/abs/1712.05520. [17] W. T. Gowers, Quasirandom groups. Combin. Probab. Comput. 17(3), 2008, 363–387. [18] H. A. Helfgott, Growth and generation in SL2 (Z/pZ). Ann. of Math. (2) 167(2), 2008, 601–623. [19] H. A. Helfgott, Growth in SL3 (Z/pZ). J. Eur. Math. Soc. 13(3) 2011, 761–851. [20] H. A. Helfgott, Growth in groups: ideas and perspectives. Bull. Amer. Math. Soc. 52(3) 2015, 357–413. [21] H. A. Helfgott. When is the union of a graph and a random permutation thereof connected? MathOverflow. https://mathoverflow.net/q/286057 (2017-11-16). ´ Seress, On the diameter of permutation groups. Ann. of Math. [22] H. A. Helfgott and A. (2) 179(2), 2014, 611–658. [23] L.-C. Kappe and R. F. Morse, On commutators in groups. In: Groups St. Andrews 2005. Vol. 2, volume 340 of London Math. Soc. Lecture Note Ser. Cambridge Univ. Press, 2007, 531–558. [24] M. W. Liebeck, On graphs whose full automorphism group is an alternating group or a finite classical group. Proc. London Math. Soc. (3) 47(2) 1983, 337–362. [25] M. W. Liebeck, On minimal degrees and base sizes of primitive permutation groups. Arch. Math. (Basel) 43(1) 1984, 11–15. [26] M. W. Liebeck and A. Shalev, Diameters of finite simple groups: sharp bounds and

Helfgott: Growth in linear algebraic groups and permutation groups

345

applications. Ann. of Math. (2) 154(2) 2001, 383–406. [27] E. M. Luks and P. McKenzie, Parallel algorithms for solvable permutation groups. J. Comput. System Sci. 37(1) 1988, 39–62. [28] A. Mar´oti, On the orders of primitive groups. J. Algebra 258(2) 2002, 631–640. [29] G. A. Miller, On the commutators of a given group. Bulletin of the American Mathematical Society 6(3) 1899, 105–109. [30] C. E. Praeger, L. Pyber, P. Spiga, and E. Szab´ o, Graphs with automorphism groups admitting composition factors of bounded rank. Proc. Amer. Math. Soc. 140(7), 2012, 2307–2318. [31] L. Pyber, On the orders of doubly transitive permutation groups, elementary estimates. J. Combin. Theory Ser. A 62(2) 1993, 361–366. [32] L. Pyber and E. Szab´o, Growth in finite simple groups of Lie type. J. Amer. Math. Soc. 29(1), 2016, 95–146. ´ Seress, Permutation group algorithms. Cambridge University Press, 2003. [33] A. [34] C. C. Sims, Computational methods in the study of permutation groups. In: Computational Problems in Abstract Algebra, Pergamon, Oxford, 1970, 169–183 [35] P. Spiga, Two local conditions on the vertex stabiliser of arc-transitive graphs and their effect on the Sylow subgroups. J. Group Theory 15(1), 2012, 23–35.

L2 -BETTI NUMBERS AND THEIR ANALOGUES IN POSITIVE CHARACTERISTIC ANDREI JAIKIN-ZAPIRAIN Departamento de Matem´ aticas, Universidad Aut´onoma de Madrid and Instituto de Ciencias Matem´aticas, CSIC-UAM-UC3M-UCM, Spain Email: [email protected]

Abstract In this article, we give a survey of results on L2 -Betti numbers and their analogues in positive characteristic. The main emphasis is made on the L¨ uck approximation conjecture and the strong Atiyah conjecture.

1

Introduction

Let G be a group and let K be a field. For every matrix A ∈ Matn×m (K[G]) and every normal subgroup N of G of finite index let us define φA G/N :

K[G/N ]n → K[G/N ]m (x1 , . . . , xn ) → (x1 , . . . , xn )A.

This is a K-linear map between two finite-dimensional K-vector spaces. Thus, we can define dimK Im φA dimK ker φA G/N G/N rkG/N (A) = =n− . (1) |G : N | |G : N | Now, let G > G1 > G2 > . . . be a descending  chain of subgroups such that Gi is normal in G, the index |G : Gi | is finite and i≥1 Gi = {1}. For a given matrix A over K[G], we want to study the sequence {rkG/Gi (A)}i≥1 . Concretely, we would like to answer the following questions. Question 1.1 Let us assume the previous notation. 1. Does the sequence {rkG/Gi (A)}i≥1 converge? 2. Assume that the limit lim rkG/Gi (A) exists. Does it depend on the chain i→∞

G > G1 > G2 > . . . ? 3. Assume that the limit lim rkG/Gi (A) exists. What are the possible values of i→∞

the limit lim rkG/Gi (A)? i→∞

These questions arise in very different situations. We will present several examples in Section 13. Let us formulate a conjecture which answers all these three questions. Conjecture 1.2 Let us assume the previous notation. Then the following holds. (1) The sequence {rkG/Gi (A)}i≥1 converges.

Jaikin-Zapirain: L2 -Betti numbers

347

(2) The limit lim rkG/Gi (A) does not depend on the chain G > G1 > G2 > . . . . i→∞

(3) Assume that there exists an upper bound for the orders of finite subgroups of G and let lcm(G) be the least common multiple of these orders. Then lim rkG/Gi (A) ∈

i→∞

1 Z. lcm(G)

Informally, the first and second part of the conjecture is called the L¨ uck approximation conjecture and the third part is called the strong Atiyah conjecture. In Section 2, we will introduce the original L¨ uck approximation and strong Atiyah conjectures. They are formulated only for fields K which are subfields of the field C of complex numbers. The numbers rkG (A) which will appear in these conjectures are generalizations of the L2 -Betti numbers invented by M. Atiyah. If K is of characteristic p > 0, then lim rkG/Gi (A) is what we call an analogue of an L2 -Betti i→∞

number in positive characteristic. If K is of characteristic 0, the parts (1) and (2) of Conjecture 1.2 are known to be true and the part (3) holds for many families of groups which include the groups from the class D, Artin’s braid groups, virtually special groups and torsion-free p-adic compact groups. If K is of positive characteristic, the parts (1) and (2) are only known when G is amenable and the part (3) when G is elementary amenable. If the reader sees Conjecture 1.2 for the first time he or she might wonder what makes the cases of characteristic 0 and positive characteristic so different. A quick answer is that in characteristic 0 we can use different techniques from the theory of operator algebras, but we do not have any analogue of them in positive characteristic. Nevertheless, in this survey we will try to give a uniform treatment of both cases using the notion of Sylvester matrix rank function. This is the main difference of our exposition of this subject from the previous ones. Our first motivation is to explain the main ideas behind the proofs of positive results concerning Conjecture 1.2 and the related conjectures. We will present the complete proofs of several results. Some of them are not new but they are formulated in the literature differently, so we think it will be convenient to include their proofs. In most cases we will give only a sketch of the proofs, providing the references where the complete proofs can be found. Another motivation is to collect together the main open problems in the area. We hope that this will stimulate further research in this subject. The article is organized as follows. In Section 2 we introduce L2 -Betti numbers of groups and formulate the strong Atiyah conjecture and different variations of the L¨ uck approximation conjecture. In Section 3 we recall basic facts about von Neumann regular and ∗-regular rings. In Section 4 we explain the notion or epic homomorphism and present the Cohn theory of epic division R-algebras. Section 5 is devoted to the theory of Sylvester matrix rank and Sylvester module rank functions. These concepts unify the notion of L2 -Betti numbers with their analogues in positive characteristic. Until now this subject has been presented in the literature only partially. Therefore, we try to describe a complete picture. We formulate several exciting questions about Sylvester rank functions. Some of them are not related to L2 -Betti numbers, but we still believe that they are of big interest. In

348

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Section 6 we give an algebraic reformulation of the conjectures described in Section 2. This algebraic point of view allows to use the techniques introduced in Sections 3, 4 and 5 in order to attack the conjectures formulated in Section 1 and 2. In Section 7 we prove the parts (1) and (2) and in Section 9 the part (3) of Conjecture 2.4 (this is a strong version of Conjecture 1.2) over an arbitrary field for amenable groups. In Section 8 we discuss the notions of natural extensions of Sylvester rank functions. They play an important role in the proofs of many results of this survey. In Section 10 we explain the proof of the general L¨ uck approximation conjecture over the field of complex numbers for sofic groups. Section 11 is devoted to the L¨ uck approximation and strong Atiyah conjecture for completed group algebras of virtually pro-p groups. We formulate questions similar to the ones from Section 1. Section 12 describes the known positive results on the strong Atiyah conjecture. Finally, in Section 13 we present several applications of Sylvester matrix rank functions and, in particular, L2 -Betti numbers in other parts of mathematics. The list of applications is far from being complete, and represents mathematical interests of the author of this survey. There are many good sources to learn about L2 -invariants and their approximations, mostly due to W. L¨ uck. First, of course, one should mention his book [78]. We also highly recommend a recent L¨ uck’s survey [81]. Other useful sources are the Ph.D. thesis of H. Reich [98], expository papers by P. Pansu [94] and B. Eckmann [29], another survey by W. L¨ uck [79] and two recent lecture notes, one by H. Kammeyer [59] and another by S. Kionke [62]. Acknowledgments This paper is partially supported by the grants MTM2017-82690-P and MTM201453810-C2-01 of the Spanish MINECO, the grant PRX16/00179 of the mobility program “Salvador de Madariaga” of the Spanish MECD and by the ICMAT Severo Ochoa project SEV-2015-0554. This article was written while the author was visiting the Mathematical Institute of the University of Oxford. I would like to thank everyone involved for their fine hospitality. I have benefited from conversations with Pere Ara, Gabor Elek, Mikhail Ershov, L

ukasz Grabowski, Rostislav Grigorchuk, Steffen Kionke, Diego L´ opez, Nikolay Nikolov, Thomas Schick, Simone Virili and Dmitry Yakubovich. I thank them sincerely. General conventions and notations In this paper all rings and homomorphisms are unital. The letter K is reserved for a field and by an algebra we will mean always a K-algebra. If R is a ring, an R-module will usually mean left R-module. The category of R-modules is denoted by R-Mod. R[x] is the ring of polynomials over R and R[x±1 ] is the ring of Laurent polynomials. A ∗-ring is a ring R with a map ∗ : R → R that is an involution (i.e., (x∗ )∗ = x, (x + y)∗ = x∗ + y ∗ , (xy)∗ = y ∗ x∗ (x, y ∈ R)). If K is a ∗-ring, then a ∗-algebra is

Jaikin-Zapirain: L2 -Betti numbers

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an algebra with an involution ∗ satisfying (λx)∗ = λ∗ x∗ (λ ∈ K, x ∈ R). An element of a ∗-ring e is called a projection if e is an idempotent (e2 = e) and e is self-adjoint (e∗ = e). If n ≥ 1 we denote by In the n by n identity matrix. For matrices A and B, A ⊕ B denotes the direct sum of A and B:   A 0 A⊕B = . 0 B For a group G, d(G) denotes the minimal number of generators of G. We denote by F(G) the set of finite subgroups of G. If there is an upper bound on the orders of finite subgroups of G, we denote by lcm(G) the least common multiple of these orders. We will write lcm(G) = ∞ if there is no such bound. For a countable set X, l2 (X) will denote the Hilbert space with Hilbertbasis the elements of X; thus l2 (X) consists of all square summable formal sums x∈X ax x with ax ∈ C and the inner product is F G   ax x, bx x = a x bx . x∈X

2

x∈X

x∈X

L2 -Betti numbers and generalizations of Conjecture 1.2

A countable group G acts by left and right multiplication on l2 (G). The right action of G on l2 (G) extends to an action of C[G] on l2 (G) and so we obtain that the group algebra C[G] acts faithfully as bounded linear operators on l2 (G). In what follows we will simply consider C[G] as a subalgebra of B(l2 (G)), the algebra of bounded linear operators on l2 (G). A finitely generated Hilbert G-module is a closed subspace V ≤ (l2 (G))n , invariant by the left action of G. A morphism between two finitely generated Hilbert G-modules U and V is a bounded G-equivariant map α : U → V . Let V ≤ (l2 (G))n be a f.g. Hilbert G-module and projV : (l2 (G))n → (l2 (G))n the orthogonal projection onto V . We put dimG V := TrG (projV ) :=

n 

projV 1i , 1i (l2 (G))n ,

i=1

where 1i is the element of (l2 (G))n having 1 in the ith entry and 0 in the rest of the entries. The number dimG V is the von Neumann dimension of V . It does not depend on the embedding of V into l2 (G)n . The reader can consult [78] where other properties of dimG V are described. Let A ∈ Matn×m (C[G]) be a matrix over C[G]. The action of A by right multi2 n 2 m plication on l2 (G)n induces a bounded linear operator φA G : (l (G)) → (l (G)) . Let us define A rkG (A) = dimG Im φA G = n − dimG ker φG . Observe that this notation is compatible with the formula (1), because if G is finite, then rkG = rkC /|G|.

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If G is a quotient of a group F and A ∈ Matn×m (C[F ]) is a matrix over C[F ], we denote by A¯ the image of A in Matn×m (C[G]). Abusing the notation, we will ¯ A ¯ write φA G for φG and rkG (A) for rkG (A). If G is not a countable group then rkG is also well defined. Take a matrix A over C[G]. Then the group elements that appear in A are contained in a finitely generated subgroup H of G. We will put rkG (A) = rkH (A). One easily checks that the value rkH (A) does not depend on the subgroup H. In [9] M. F. Atiyah introduced for a closed Riemannian manifold (M, g) with uni˜ the analytic L2 -Betti numbers b(2) versal covering M p (M, g) which measure the size ˜ . J. Dodziuk [24] extended of the space of harmonic square-integrable p-forms on M 2 the notion of L -Betti numbers to the more general context of free cocompact actions of discrete groups on CW -complexes. In particular, he also showed that the analytic L2 -Betti numbers do not depend on the metric. For a given subfield K of C we denote by CK (G) the set of possible values rkG (A) where A is a matrix over K[G] and by AK (G) the additive group generated by CK (G). Over time it has been realized (see [29, Proposition 3.10.1]) that L2 -Betti numbers, arising from a given group G acting freely and cocompactly on CW -complexes, form a set that can be defined purely in terms of G, without mentioning CW -complexes. In our notation it is the set CQ (G). In this survey we will consider not only CQ (G) but also the sets CK (G) where K is an arbitrary subfield of C. 2.1

Atiyah’s question and the general Atiyah problem

In [9, page 72] M. F. Atiyah asked whether L2 -Betti numbers of a closed manifold can be irrational. We reformulate this question as the following problem and we refer to it as the general Atiyah problem for G. Problem 2.1 For a given group G and a given subfield K of C determine the group AK (G). Before the work of R. Grigorchuk and A. Zuk [50], it had been conjectured that G F 1 :H≤G . AQ (G) = |H| However, in [50] the authors showed that if G = C2  Z is the lamplighter group, then 1/3 ∈ AQ (G). Observe that the finite subgroups of the lamplighter group have orders which are powers of 2. This result was used in [48] to produce a closed Riemannian manifold (M, g) of dimension 7 with π1 (M ) having only finite (2) subgroups of order a power of 2 and such that b3 (M, g) = 1/3. Shortly afterwards W. Dicks and T. Schick described in [22] an element T from the group ring of Z[G] where G = (C2  Z) × (C2  Z) such that rkG (T ) looked like an irrational number. The question of irrationality of that specific number remains open. This was the first evidence that the question of Atiyah has an affirmative answer. It was T. Austin [10] who first proved the existence of a group G with an irrational element in CQ (G). His construction was not explicit. Concrete examples

Jaikin-Zapirain: L2 -Betti numbers

351

appear in [42, 67, 97, 44]. These examples also leaded to constructions of closed Riemannian manifolds with irrational L2 -Betti numbers confirming the prediction of M. Atiyah. Moreover, in [42] L

. Grabowski showed that any non-negative real number belongs to CQ (G) for some elementary amenable group G and the set of L2 -Betti numbers arising from finitely presented groups contains the set of all numbers with computable binary expansions. All the previous examples involve groups having finite subgroups of unbounded order. This suggests that we have to consider the general Atiyah problem for groups with bounded orders of finite subgroups and, in particular, for torsion-free groups. 2.2

The strong Atiyah conjecture

Now let us state a conjecture that got the name of the strong Atiyah conjecture [78]. Conjecture 2.2 (The strong Atiyah conjecture over K for a group G) Let G be a group and let K be a subfield of C. Assume that lcm(G) < ∞. Then AK (G) =

1 Z= lcm(G)

F

G 1 :H≤G . |H|

There is a considerable body of work to establish the strong Atiyah conjecture for suitable classes of groups and fields. We will present these results in Section 12. At this moment the conjecture is known over C for many families of groups which include the groups from the class D, Artin’s braid groups, virtually special groups and torsion-free p-adic compact groups. 2.3

The L¨ uck approximation conjecture

Now we introduce the L¨ uck approximation conjecture. It arised from a question of M. Gromov (which was solved by W. L¨ uck in [76]) of whether L2 -Betti numbers of a compact Riemannian manifold can be approximated by ordinary normalized Betti numbers of finite covers of the manifold. Conjecture 2.3 (The L¨ uck approximation conjecture over K for a group G) Let K be a subfield of C, F a finitely generated free groupand F > N1 > N2 > . . . a chain of normal subgroups of F with intersection N = Ni . Put Gi = F/Ni and G = F/N . Then for every A ∈ Matn×m (K[F ]), lim rkGk (A) = rkG (A).

k→∞

This conjecture was formulated by W. L¨ uck. When K is of characteristic 0, Conjecture 2.3 implies the first and the second part of Conjecture 1.2 and Conjecture 2.2 and Conjecture 2.3 together imply the third part of Conjecture 1.2.

Jaikin-Zapirain: L2 -Betti numbers

352 2.4

The sofic L¨ uck approximation conjecture

Let F be a free finitely generated group and assume that it is freely generated by a set S. Recall that an element w of F has length n if w can be expressed as a product of n elements from S ∪ S −1 and n is the smallest number with this property. Bk (1F ) will denote the set of elements of F of length at most k. Let N be a normal subgroup of F . We put G = F/N . We say that G is sofic if there is a family {Xk }k∈N of finite F -sets (F acts on the right) such that if we put Tk,s = {x ∈ Xk : x = x · w if w ∈ Bs (1F ) ∩ N, and x = x · w if w ∈ Bs (1F ) \ N }, then for every s, lim

k→∞

|Tk,s | = 1. |Xk |

The family of F -sets {Xk } is called a sofic approximation of G. This is one of many equivalent definitions of soficity for a finitely generated group; we have borrowed this one from [109, Proposition 1.4]. This definition has the following geometric meaning. The action of F on Xk  converts Xk in an S ±1 -labeled graph. Let Tk,s be the set of vertices x of Xk such that the s-ball Bs (x) in Xk and the s-ball Bs (1G ) in G are isomorphic as S ±1 -labeled graphs. It is clear that   ⊆ Tk,s ⊆ Tk,2s . Tk,s

Thus, the soficity condition says that for every s most of the vertices of Xk are in  when k tends to infinity. Tk,s For an arbitrary group G we say that G is sofic if every finitely generated subgroup of G is sofic. Amenable groups and residually finite groups are sofic. It is important to note that no nonsofic group is known at this moment. On the other hand, all the results presented in this survey are about sofic groups. Now, let us generalize slightly the notation introduced in Section 1. Let F be a group acting (on the right) on a finite set X and let K be a field. For every matrix A ∈ Matn×m (K[F ]) let us define φA X :

K[X]n → K[X]m . (x1 , . . . , xn ) → (x1 , . . . , xn )A

This is a K-linear map between two finite-dimensional K-vector spaces, and so, we can define dimK ker φA dimK Im φA X X rkX (A) = =n− . (2) |X| |X| Conjecture 2.4 (The sofic L¨ uck approximation conjecture over K for a group G) Let {Xk } be a sofic approximation of G = F/N . Then (1) for every A ∈ Matn×m (K[F ]), there exists the limit lim rkXk (A); k→∞

(2) the limit does not depend on the sofic approximation {Xi };

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(3) If K is a subfield of C, then lim rkXk (A) = rkG (A). k→∞

This conjecture generalizes the parts (1) and (2) of Conjecture 1.2. Conjecture 2.4 holds when K is of characteristic 0. When K is of positive characteristic, the first and second parts of Conjecture 2.4 hold when G is amenable. These results will be explained in Sections 7 and 10. 2.5

The L¨ uck approximation in the space of marked groups.

Let F be a free group freely generated by a finite set S. The space of marked groups MG(F, S) can be identified with the set of normal subgroups of F with the metric d(N1 , N2 ) = e−n where n is the largest integer such that the balls of radius n in the Cayley graphs of F/N1 and F/N2 with respect to the generators S are simplicially isomorphic (with respect to an isomorphism respecting the labelings). In this setting the approximation conjecture is stated in the following way. Conjecture 2.5 (The L¨ uck approximation conjecture in the space of marked groups over K for a group G) Let K be a subfield of C. Let {Nk ∈ MG(F, S)} converge to N ∈ MG(F, S). Put G = F/N and Gk = F/Nk . Then for every A ∈ Matn×m (K[F ]), lim rkGk (A) = rkG (A).

k→∞

Clearly Conjecture 2.5 is a strong version of Conjecture 2.3. It is known in the case where the groups Gi are sofic. This will be a part of a more general conjecture which we discuss in the next subsection. 2.6

The general L¨ uck approximation conjecture

In this subsection we will introduce a new type of approximation that unify together the sofic approximation and the approximation in the space of marked groups. Then we will formulate the L¨ uck approximation conjecture for this general situation. As before, let F be a finitely generated free group, freely generated by a finite set S, N a normal subgroup of F and G = F/N . Let {Hk }k∈N be a family of groups and Xk an (Hk , F )-set (i.e., Hk acts on the left, F acts on the right and these two actions commute) such that Hk acts freely on Xk and Hk \Xk is finite. We define Tk,s = {x ∈ Xk : x = x · w if w ∈ Bs (1F ) ∩ N, and x = x · w if w ∈ Bs (1F ) \ N }. Then we say that {Xk } approximates G if for every s, lim

k→∞

|Hk \Tk,s | = 1. |Hk \Xk |

The sofic approximation is a particular case of the general approximation and corresponds to the case when the groups Hk are trivial. The approximation in the space of marked groups arises from the general approximation in the case when Hk and F act transitively on Xk for every k.

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As in the case of sofic approximation, the general approximation has a geometric interpretation. We see Xk as an S ±1 -labeled graph. Since the action of Hk and F commutes, the elements of Hk act on Xk as S ±1 -labeled graph isomorphisms. Therefore, for every s ∈ N the balls of radius s centered in the vertices of an Hk orbit in Xk are isomorphic. There are only finitely many Hk -orbits in Xk and the approximation condition says that when k tends to infinity, for almost all of them, the ball of radius s centered in a point of the orbit is isomorphic to Bs (1G ). Now, we can generalize the previous notation in the following way. Let A ∈ Matn×m (C[F ]) be a matrix over C[F ]. Let H be a group and let X be an (H, F )set such that H acts freely on X and H\X is finite. By multiplication on the right 2 n 2 m side, A induces a linear operator φA X : (l (X)) → (l (X)) . We put rkX (A) =

dimH Im φA dimH ker φA X X =n− . |H\X| |H\X|

Conjecture 2.6 (The general L¨ uck approximation conjecture over K for a group G) Let K be a subfield of C, F a finitely generated free group and N a normal subgroup of F . For each natural number k, let Xk be an (Hk , F )-set such that Hk is a group that acts freely on Xk and Hk \Xk is finite. Assume that {Xk } approximates G = F/N . Then for every A ∈ Matn×m (K[F ]), lim rkXk (A) = rkG (A).

k→∞

This conjecture generalizes all the previous variations of the L¨ uck approximation conjecture over fields of characteristic 0. We will explain in Section 10 the proof of this conjecture over C in the case where all groups Hk are sofic. If G is an arbitrary group, we say that G satisfies the general L¨ uck approximation conjecture over K if all its finitely generated subgroups do.

Von Neumann regular and ∗-regular rings

3 3.1

Von Neumann regular rings

An element x of a ring R is called von Neumann regular if there exists y ∈ R satisfying xyx = x. A ring U is called von Neumann regular if all the elements of U are von Neumann regular. In the following proposition we collect the properties of von Neumann regular rings that we will need later. Proposition 3.1 ([46]) Let U be a von Neumann regular ring. Then the following statements hold: 1. every finitely generated left ideal of U is generated by an idempotent; 2. every finitely generated left submodule of a projective module P of U is a direct summand of P (and, in particular, it is projective); 3. every finitely generated left projective module of U is a direct sum of left cyclic ideals of U .

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The ring of unbounded affiliated operators of a group

The ring of unbounded affiliated operators U (G) of a group G is one of the main examples of a von Neumann regular ring that appear in this survey. The Ph.D. thesis of H. Reich [98] is a good source to learn basic facts about the ring U(G). We briefly define it in this subsection and also introduce additional notions that will motivate further definitions. Let G be a countable group. The group von Neumann algebra N (G) of G is the algebra of G-equivariant bounded operators on l2 (G): N (G) = {φ ∈ B(l2 (G)) : φ(gv) = gφ(v) for all g ∈ G, v ∈ l2 (G)}. It can be defined also as the weak closure of C[G] in B(l2 (G)) or, algebraically, as the second centralizer of C[G] in B(l2 (G)). The ring N (G) satisfies the left Ore condition (a result proved by S. K. Berberian in [13]). We recall this notion in Subsection 4.1. The left classical ring of fractions Ql (N (G)) is denoted by U (G). The ring U (G) can be also described as the ring of densely defined (unbounded) operators which commute with the left action of G. Therefore, U (G) is called the ring of unbounded affiliated operators of G. The ring U (G) is a ∗-regular ring. We will consider such rings in more detail in Subsection 3.4. We can define a rank function rkG on U (G) in the following way rkG (s−1 r) = rkG (r) = dimG (l2 (G)r) = projl2 (G)r 1, 1 l2 (G) ,

(3)

where r ∈ N (G) and s ∈ N (G) is a non-zero-divisor in N (G). Note that if u ∈ U(G), then rkG (u) = 1 if and only if u is invertible in U (G).

(4)

The function rkG can be extended to all matrices over U (G) and it is an example of a faithful Sylvester matrix rank function on a ∗-regular ring. We will consider the Sylvester rank functions in more detail in Section 5. The Sylvester matrix rank function rkG induces a Sylvester module rank function dimG on finitely presented left modules of U (G) (see Subsection 5.3 for more details) that satisfies dimG (U (G)u) = rkG (u), u ∈ U(G). 3.3

Von Neumann regular elements in a proper ∗-ring

Let R be a ∗-ring. The involution  ∗ is called proper if x∗ x = 0 implies x = 0 and it is called n-positive definite if ni=1 x∗i xi = 0 implies x1 = · · · = xn = 0. Thus, the involution is proper if and only if it is 1-positive definite. If the involution is n-positive definite for all n, then we say that it is positive definite. We say that a ∗-ring is proper if its involution is proper. In general if x is a von Neumann regular element there are several elements y satisfying xyx = x. However, if R is a proper ∗-ring there is a canonical one. In the following proposition we collect the main properties of regular elements in a proper ∗-ring.

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Proposition 3.2 ([46, 56]) Let R be a proper ∗-ring and let x ∈ R. Assume that x∗ x and xx∗ are von Neumann regular elements. Then the following holds. 1. Rx = Rx∗ x. 2. x and x∗ are von Neumann regular. 3. There exists a unique projection e in R such that Re = Rx and there exists a unique projection f such that f R = xR (we put e = RP(x) and f = LP(x)). 4. There exists a unique y ∈ eRf such that yx = e and xy = f (we put x[−1] = y and call it the relative inverse of x). 5. RP(x) = RP(x∗ x) = LP(x) and (x∗ )[−1] = (x[−1] )∗ . 6. (x∗ x)[−1] = x[−1] (x∗ )[−1] and x[−1] = (x∗ x)[−1] x∗ . 7. If x is self-adjoint, then x commutes with x[−1] . 3.4

Von Neumann ∗-regular rings

A ∗-ring U is called von Neumann ∗-regular (or simply ∗-regular) if it is von Neumann regular and its involution is proper. The ring Matn (C) is ∗-regular. The ring C[G] is ∗-regular if and only if G is locally finite. However, we can embed C[G] in the ∗-regular ring U (G) for an arbitrary group G. A direct product of ∗-regular rings is again ∗-regular. If U is a ∗-regular ring, then Matn (U ) is again a ∗-ring: if M = (mij ) then M ∗ = (nij ) with nij = (m∗ji ). Also Matn (U ) is von Neumann regular. However, in general ∗ is not proper in Matn (U ). We say that U is a positive definite ∗-regular if Matn (U ) is ∗-regular for every n ∈ N. It is equivalent to the condition that ∗ is positive definite. For example, Matn (C) and U (G) are positive definite ∗-regular rings. Although in the definition of a ∗-regular ring the properties to be von Neumann regular and to be proper do not interact, using them together we obtain many interesting consequences. For example, if I is an ideal of a ∗-regular ring U , then I is automatically ∗-invariant and moreover ∗ induces a proper involution on U /I. The following proposition explains how to construct the minimal ∗-regular subring containing a given ∗-subring. This was proved first for positive definite ∗regular rings by P. Linnell and T. Schick in [72] and by P. Ara and K. Goodearl in the form that we present here in [7, Proposition 6.2]. Let R be a ∗-subring of a ∗-regular ring U . We denote by R1 (R, U ) the subring of U generated by R and all the relative inverses of all the elements x ∈ R. Clearly R1 (R, U ) is again a ∗-subring of U . We put Rn+1 (R, U ) = R1 (Rn (R, U ), U ). Proposition 3.3 ([7, Proposition 6.2]) Let U be a ∗-regular ring and let R be a ∗-subring of U . Then there is a smallest ∗-regular subring R(R, U ) of U containing R. Moreover, ∞  R(R, U ) = Ri (R, U ). i=1

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The subring R(R, U ) is called the ∗-regular closure of R in U . It was observed in [56] that, in fact, R1 (R, U ) can be also defined as the subring of U generated by R and all the relative inverses of the elements of the form x∗ x for x ∈ R. If K is a subfield of C closed under complex conjugation and G is a countable group, then the ∗-regular closure of K[G] in U(G) is denoted by RK[G] . For an arbitrary group G, RK[G] is defined as the direct union of {RK[H] : H is a finitely generated subgroup of G}.

The Cohn theory of epic division R-algebras

4 4.1

The Ore localization

In this subsection we recall the definition of the left Ore condition and the construction of the Ore ring of fractions. An element r ∈ R is a non-zero-divisor if there exists no non-zero element s ∈ R such that rs = 0 or sr = 0. Let T be a multiplicative subset of non-zerodivisors of R. We say that (T, R) satisfies the left Ore condition if for every r ∈ R and every t ∈ T , the intersection T r ∩ Rt is not trivial. If T consists of all the non-zero-divisors we simply say that R satisfies the left Ore condition. The goal is to construct the left Ore ring of fractions T −1 R. Let us recall briefly this construction. For more details the reader may consult [84, Chapter 2]. As a set, T −1 R coincides with the set of equivalence classes in T × R with respect to the following equivalence relation: (t1 , r1 ) ≡ (t2 , r2 ) if and only if there are r1 , r2 ∈ R such that r1 t1 = r2 t2 ∈ T and r1 r1 = r2 r2 . The equivalence class of (t, a) is denoted by t−1 a. Note that there is no obvious interpretation for the sum s−1 a + t−1 r and the product (t−1 r)(s−1 a) (a, r ∈ R, s, t ∈ T ). In order to sum s−1 a and t−1 r, we observe that for every s, t ∈ T there exists s , t ∈ R such that s s = t t ∈ T . Hence, s−1 a + t−1 r = (s s)−1 s a + (t t)−1 t r = (s s)−1 (s a + t r) In order to multiply s−1 a and t−1 r, we rewrite rs−1 as a product (s0 )−1 r0 with r0 ∈ R and s0 ∈ T . The condition T r∩Rs is not trivial implies exactly the existence of s0 ∈ T and r0 ∈ R such that s0 r = r0 s, and so rs−1 = (s0 )−1 r0 . Hence, (t−1 r)(s−1 a) = (t−1 )(s0 )−1 r0 a = (s0 t)−1 r0 a. When T consists of all the non-zero-divisors of R and (T, R) satisfies the left Ore condition, we denote T −1 R by Ql (R) and we call it the left classical ring of fractions of R. An important result in the theory of classical rings of quotients is Goldie’s theorem [47, Theorem 6.15]. One of its consequences (see [47, Corollary 6.16]) is that every semiprime left Noetherian ring has a semisimple Artinian classical left ring of fractions.

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Rational closure

Let R be a subring of S. Denote by GL(R; S) the set of square matrices over R which are invertible over S. The rational closure of R in S is the subring of S generated by all the entries of the matrices M −1 for M ∈ GL(R; S) (in fact, the entries of the matrices M −1 for M ∈ GL(R; S), form a subring). Let f : R → S be a map and let Σ be a set of matrices over R such that f (Σ) ⊂ GL(f (R); S). Then there exists the universal localization of R with respect to Σ. It is an R-ring λ : R → RΣ such that every element from λ(Σ) is invertible over RΣ and every Σ-inverting homomorphism from R to another ring can be factorized uniquely by λ (see [19, Theorem 4.1.3]). An Ore localization is a particular case of universal localization. A useful result to study rational clousures is Cramer’s rule ([19, Proposition 4.2.3], [20, Proposition 7.1.5]). One of its consequences is the following proposition. Proposition 4.1 Let S be a rational closure of R. Then for every matrix A over S there are k ≥ 1, a matrix A over R and matrices P and Q which are invertible over S such that A ⊕ Ik = P A Q. 4.3

Epic homomorphisms

Let f : R → S be a ring homomorphism. We say that f is epic if for every ring Q and homomorphisms α, β : S → Q, the equality α ◦ f = β ◦ f implies α = β. An epic R-ring is a pair (S, f ) where f : R → S is epic. For simplicity we will write S instead of (S, f ) when f is clear from the context. For example, if S is the rational closure of f (R) in S, then f is epic. We will say that two epic R-rings (S1 , f1 ) and (S2 , f2 ) are isomorphic if there exists an isomorphism α : S1 → S2 for which the following diagram is commutative: Id

R −−−−→ ⏐ ⏐f .1

R ⏐ ⏐f .2

α

S1 −−−−→ S2 Epic homomorphisms can be characterized in the following way. Proposition 4.2 ([107, Proposition XI.1.2]) Let f : R → S be a ring homomorphism. Then f is epic if and only if the multiplication map m : S ⊗R S → S is an isomorphism of S-bimodules. More generally if f : R → S is a ring homomorphism, we say that s ∈ S is dominated by f if for any ring Q and homomorphisms α, β : S → Q, the equality α ◦ f = β ◦ f implies α(s) = β(s). The set of elements of S dominated by f is a subring of S, called the dominion of f . The following result implies that an epic homomorphism from a von Neumann regular ring is always surjective.

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Proposition 4.3 ([107, Proposition XI.1.4]) Let U be a von Neumann regular ring. Then for every ring homomorphism γ : U → S, the dominion of γ is equal to γ(U ). 4.4

A characterization of epic division R-rings

An epic division R-ring is an epic R-ring f : R → D, where D is a division ring. Applying Proposition 4.3, it is not difficult to see that for an epic division R-ring (D, f ), D is the rational closure of f (R) in D. If R is a commutative ring, then there exists a natural bijection between Spec(R) and the isomorphism classes of division R-rings: a prime ideal P ∈ Spec(R) corresponds to the field of fractions Q(R/P ) of R/P and f : R → Q(R/P ) is defined as f (r) = r + P for any r ∈ R. If R is a domain and satisfies the left Ore condition then its classical left ring of fractions Ql (R) is a division ring. Moreover, as in the commutative case, the division R-ring Ql (R) is the unique (up to R-isomorphism) faithful division R-ring. Thus, if R is a left Noetherian ring, then there exists a natural bijection between the strong prime ideals of R (ideals P such that R/P is a domain) and the isomorphism classes of division R-rings. For an arbitrary ring R, P. Cohn proposed the following approach to classify division R-rings. If D is a division ring, let rkD (M ) be the D-rank of a matrix M over D. Theorem 4.4 ([19, Theorem 4.4.1]) Let (D1 , f1 ) and (D2 , f2 ) be two epic division R-rings. Then (D1 , f1 ) and (D2 , f2 ) are isomorphic if and only if for each matrix M over R rkD1 (f1 (M )) = rkD2 (f2 (M )).

5

Sylvester rank functions

The functions rkG/N and rkX which have appeared in Sections 1 and 2 are examples of Sylvester matrix rank functions on the algebra K[G]. In this section we introduce the notion of Sylvester rank functions on an arbitrary algebra and study their properties. 5.1

Sylvester matrix rank functions

Let R be an algebra. A Sylvester matrix rank function rk on R is a function that assigns a non-negative real number to each matrix over R and satisfies the following conditions. (SMat1) rk(M ) = 0 if M is any zero matrix and rk(1) = 1; (SMat2) rk(M1 M2 ) ≤ min{rk(M1 ), rk(M2 )} for any matrices M1 and M2 which can be multiplied;   M1 0 = rk(M1 ) + rk(M2 ) for any matrices M1 and M2 ; (SMat3) rk 0 M2

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M1 M3 0 M2 appropriate sizes.

(SMat4) rk

 ≥ rk(M1 ) + rk(M2 ) for any matrices M1 , M2 and M3 of

If φ : F1 → F2 is an R-homomorphism between two free finitely generated Rmodules F1 and F2 , then rk(φ) is rk(A) where A is the matrix associated with φ with respect to some R-bases of F1 and F2 . It is clear that rk(φ) does not depend on the choice of the bases. The following elementary properties of a Sylvester matrix rank function can be obtained from its definition. Proposition 5.1 Let R be an algebra and let rk be a Sylvester matrix rank function on R. Let A, B ∈ Matn×m (R), and C ∈ Matm×k (R). Then 1. rk(A + B) ≤ rk(A) + rk(B). 2. rk(AC) ≥ rk(A) + rk(C) − m. Proof The first statement is proved in [56]. Let us show (2). Indeed, we have that   AC 0n×m SMat3 rk(AC) + m = rk 0m×k Im     SMat2 Ik 0k×m AC 0n×m 0m×n −Im ≥ rk In A 0m×k Im −C Im   SMat4 C −Im ≥ rk(A) + rk(C). = rk 0n×k A  For any algebra R we denote by P(R) the set of the Sylvester matrix rank functions on R. The set P(R) is a compact convex subset of functions on matrices over R (with respect to the point convergence topology). It is hard to calculate P(R) for a general algebra R (see [57] where various examples of explicit calculations of P(R) are presented). For a given homomorphism f : R → S of algebras, we define f # : P(S) → P(R) by f # (rk)(M ) = rk(f (M )), where M is a matrix over R. 5.2

Sylvester matrix rank functions and rational closures

Proposition 5.2 Let f : R → S be a homomorphism of algebras. Assume that S is a rational closure of f (R). Then f # is injective. Moreover, if S = RΣ is a universal localization, then Im f # = {rk ∈ P(R) : rk(A) = n if A ∈ Σ ∩ Matn (R)}. In particular, if T is a multiplicative set of non-zero-divisors of R, (T, R) satisfies the left Ore condition and S = T −1 R, then Im f # = {rk ∈ P(R) : rk(t) = 1 for all t ∈ T }.

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Proof The first part of the proposition follows from Proposition 4.1 and the second one is proved in [105, Theorem 7.4]. The proof of [105, Theorem 7.4] is quite technical. Let us present here the proof of the last statement of the proposition, which will also give an idea about the proof of the general case. Let rk ∈ P(R) be such that rk(t) = 1 for all t ∈ T . We want to extend rk on T −1 R. Given A = t−1 B ∈ (t ∈ T , B a matrix over R) we put rk(A) = rk(B). The main difficulty is to show that this definition does not depend on the choice of −1 the pair (t, B). Assume that we can write A also as t−1 1 B1 (t1 ∈ T , B1 a matrix over R). We have to show that rk(B) = rk(B1 ). Applying the definition of Ore condition, we obtain that there are a, b ∈ R such that at = bt2 ∈ T and aB = bB1 . Since rk(at) = rk(bt2 ) = 1, we have that rk(a) = rk(b) = 1. Hence rk(B)

Proposition 5.1(2)

=

rk(aB) = rk(bB1 )

Proposition 5.1(2)

=

rk(B1 ).

Thus, the extension of rk on T −1 R is well-defined. Now, it is not difficult to see that it is indeed a Sylvester matrix rank function on T −1 R.  In view of this proposition, we will identify P(RΣ ) with the corresponding subset of P(R). 5.3

Sylvester module rank functions

A Sylvester module rank function dim is a function that assigns a non-negative real number to each finitely presented R-module and satisfies the following conditions. (SMod1) dim({0}) = 0, dim(R) = 1; (SMod2) dim(M1 ⊕ M2 ) = dim M1 + dim M2 ; (SMod3) if M1 → M2 → M3 → 0 is exact then dim M1 + dim M3 ≥ dim M2 ≥ dim M3 . Given a matrix A ∈ Matn×m (R) we put MA = Rm /(Rn )A. It is clear that MA is a finitely presented left R-module. Conversely, given a finitely presented left R-module M we can find a matrix A ∈ Matn×m (R) such that MA ∼ = M . This observation allows to construct a natural one-to-one correspondence between the Sylvester matrix rank functions and the Sylvester module rank functions. Proposition 5.3 ([83],[105, Chapter 7]) Let R be an algebra. 1. Let rk be a Sylvester matrix rank function on R and let A ∈ Matn×m (R). We put dim(MA ) = m − rk(A). Then dim is well defined and it is a Sylvester module rank function on R. 2. Let dim be a Sylvester module rank function on R and let A ∈ Matn×m (R). We put rk(A) = m − dim(MA ). Then rk is a Sylvester module rank function on R.

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If rk and dim are related as described in Proposition 5.3 we will say that they are associated. 5.4

The pseudo-metric induced by a Sylvester matrix rank function

Given a Sylvester matrix rank function rk on R, we define δ(x, y) = rk(x − y), x, y ∈ R. Proposition 5.1(1) implies that the function δ is a pseudo-metric on R. Even though δ is not always a metric, we refer to it as rk-metric for convenient abbreviation. Observe that the set ker rk = {a ∈ R : rk(a) = 0} is an ideal of R. We say that rk is faithful if ker rk = 0. By Proposition 5.1(1), rk may be seen as a faithful Sylvester matrix rank function on the quotient ring R/ ker rk, and so, δ is a metric on R/ ker rk. Since the multiplication and addition on R are uniformly continuous with respect to δ, the (Hausdorff) completion of R/ ker rk, which we denote by Rrk (or simply R when rk is clear from the context) is a ring. The kernel of the natural map R → Rrk is ker rk. The function rk can be extended by continuity on Rrk and on matrices over Rrk and one easily may check that this extension (denoted also by rk) is a Sylvester matrix rank function on Rrk . If G is a group and K a subfield of C, then the completion of RK[G] with respect to the rkG -metric is denoted by RK[G] . 5.5

Exact Sylvester rank functions

We say that a Sylvester module rank function dim on R is exact if it satisfies the following condition (SMod3 ) given a surjection φ : M N between two finitely presented Rmodules, dim M − dim N = inf{dim L : L ker φ and L is finitely presented}. The following result is proved by S. Virili in [112]. Proposition 5.4 ([112]) Let R be an algebra and let dim be an exact Sylvester module rank function on R. For every finitely generated R-module M put dim M = inf{dim L : L M and L is finitely presented}, and for every arbitrary R-module put (LF 1) dim M = sup{dim L : L ≤ M and L is finitely generated}. Then the extended function dim : R-Mod → R≥0 ∪ {+∞} satisfies the following condition. (LF2) if 0 → M1 → M2 → M3 → 0 is exact then dim M1 + dim M3 = dim M2 .

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A function on R-Mod satisfying (LF1) and (LF2) is called a length function. If a length function l satisfies l(R) = 1, then the restriction of l on finitely presented R-modules is an exact Sylvester module rank function on R. Moreover, l can be recovered from this restriction using the formulas which appear in Proposition 5.4. Length functions were first considered by D. Northcott and M. Reufel [90], generalizing the composition length of modules. This concept was investigated later by P. V´amos [111]. For more recent results the reader may consult [100, 113] and references therein. Note that the most interesting examples of length functions l on an algebra R do not satisfy the condition l(R) is finite, and so, do not induce Sylvester module rank functions on R. Thus, the theory of length functions is almost parallel to the theory of Sylvester module rank functions. 5.6

Sylvester rank functions on von Neumann regular rings

An arbitrary algebra may not have an exact Sylvester module rank function. However, if U is von Neumann regular, then, by Proposition 3.1(2), finitely presented U -modules are projective, and so, all the exact sequences of finitely presented U modules split. Thus, every Sylvester module rank function on a regular algebra U is exact. Note also that, by Proposition 3.1(3), a Sylvester matrix rank function on a von Neumann regular algebra U is completely determined uniquely by its values on elements from U . Thus, pseudo-rank functions studied in [46] are exactly our Sylvester matrix rank functions. Let us mention one result from this book. Proposition 5.5 ([46]) Let U be a von Neumann regular algebra and rk a Sylvester matrix rank function. 1. The algebra Urk is also von Neumann regular. 2. The following conditions are equivalent: (a) Z(Urk ) is a field; (b) Urk is simple; (c) rk is the only Sylvester matrix rank function on Urk . The conditions of the previous proposition hold in the following example. Recall that a group G is called ICC group if all the non-trivial conjugacy classes of G are infinite. Proposition 5.6 ([56]) Let G be an ICC group and K a subfield of C closed under complex conjugation. Then Z(RK[G] ) is a subfield of C. We finish this subsection with the following definition. A Sylvester matrix rank function rk on an arbitrary algebra R is called regular if there exists an algebra homomorphism f : R → U such that U is von Neumann regular and rk ∈ Im f # . In this case U is called a regular envelope of rk. Clearly, rk may have many regular envelopes. Later we will see that in some cases we can speak about the canonical regular envelope attached to rk. At this moment all known examples of Sylvester rank functions on an algebra are regular.

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Question 5.7 Let R be an algebra. Is it true that every Sylvester rank function on R is regular? 5.7

Ultraproducts of von Neumann regular rings

Given a set X, an ultrafilter on X is a set ω consisting of subsets of X such that 1. the empty set is not an element of ω; 2. if A and B are subsets of X, A is a subset of B, and A is an element of ω, then B is also an element of ω; 3. if A and B are elements of ω, then so is the intersection of A and B; 4. if A is a subset of X, then either A or X \ A is an element of ω. If a ∈ X, we can define ωa = {A ⊆ X : a ∈ A}. It is a ultrafilter, called a principal ultrafilter. It is a known fact that if X is infinite, then the axiom of choice implies the existence of a non-principal ultrafilter. Let ω be a ultrafilter on X and {ai ∈ R}i∈X a family of real numbers. We write a = lim ai if for any  > 0 the set {i ∈ X : |a − ai | < } is an element of the ω

ultrafilter w. It is not difficult to see that for any bounded family {ai ∈ R}i∈X there exists a unique a ∈ R such that a = lim ai . ω

Now, let {Ui }i∈X be a family of von Neumann regular rings and for each i ∈ X let rki be a Sylvester matrix rank function on Ui . Then i∈X Ui is a von Neumann regular ring. Let ω be an ultrafilter on X. We put  rkω (r) = lim rki (ri ), where r = (r1 , r2 , . . .) ∈ Ui . ω

i∈X

One easily obtains that rkω is a Sylvester matrix rank function on define   Ui = ( Ui )/ ker(rkω ). 

ω

 i∈X

Ui . We

i∈X

von Neumnn regular ring and rkω is a faithful Sylvester matrix Then ω Ui is a  rank function on ω Ui . For an algebra R, we denote by Preg (R) the space of regular Sylvester matrix rank functions on R. The previous construction implies the following proposition. Proposition 5.8 ([56]) Preg (R) is a closed convex subset of P(R). 5.8

Sylvester rank functions on epic von Neumann regular R-rings

Let R be an algebra and let f : R → U be an epic von Neumann regular R-ring. From the following proposition, proved in [56], we obtain that any Sylvester matrix rank function on U is completely determined by its values on matrices over f (R). Proposition 5.9 ([56]) Let R be a subalgebra of a von Neumann regular algebra U . Assume that the embedding of R in U is epic. Then for any r1 , . . . rk ∈ U, there

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is a matrix M of size a × b over R and there are vectors v1 , . . . , vk ∈ Rb such that for every t1 , . . . , tk ∈ R and every Sylvester matrix rank function rk on U ,   M rk(t1 r1 + . . . + tk rk ) = rk − rk(M ). t1 v 1 + . . . + tk v k This proposition can be applied, for example, in the case where U is a division algebra. But in this case it follows already from Proposition 4.1. Another interesting application of this proposition is presented in Subsection 5.10. In the proof of Proposition 5.9 the condition that U is regular plays an important role. Nevertheless, we want to raise the following question. Question 5.10 Let f : R → S be an epic homomorphism between two algebras. Is it true that the map f # : P(S) → P(R) is injective? If S is a rational closure of R, then a positive answer on the previous question follows from Proposition 5.2. Proposition 5.9 suggests that if R is an algebra and rk is a Sylvester matrix rank function on R having an epic von Neumann regular envelope, then this envelope might be “canonical”. As we have seen this happens in the case where the envelopes are division algebras. We formulate this precisely as the following question. Question 5.11 Let rk be a Sylvester matrix rank function on R having two epic von Neumann regular envelopes U1 and U2 . Is it true that U1 and U2 are isomorphic as R-rings? More generally, let U be another von Neumann regular envelope for rk. Is there an R-homomorphism f : U1 → U? As we have mentioned before, the answer to both questions is positive if U1 is a division algebra. 5.9

Sylvester matrix rank functions on ∗-regular rings

Now consider Sylvester rank functions on ∗-regular rings. In the following proposition we see that a Sylvester matrix rank function on a ∗-regular ring is always ∗-invariant. Proposition 5.12 Let rk be a Sylvester matrix rank function on a ∗-regular ring U and M ∈ Matn×m (U ). Then rk(M ) = rk(M ∗ ). Proof Without loss of generality we may assume that n = m and M ∈ Matn (U ). It is clear that if a, b ∈ U and aU = bU or U a = U b, then rk(a) = rk(b). Hence for every r ∈ U, rk(r) = rk(RP(r)) = rk(LP(r∗ )) = rk(r∗ ). Observe that the function rk∗ defined as rk∗ (X) = rk(X ∗ ), X is a matrix over R,

(5)

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is also a Sylvester matrix rank function on U. We want to show that rk = rk∗ . This will follow immediately if we show that the Sylvester module rank functions dim and dim∗ associated with rk and rk∗ respectively, defined as in Proposition 5.3, coincide. Note that if X ∈ Matn×m (U), then rk(X) = m − dim(U m /U n · X) = dim (U n · X) and rk∗ (X) = m − dim∗ (U m /U n · X) = dim∗ (U n · X). Now, from (5) we obtain that dim(U r) = rk(r) = rk∗ (r) = dim∗ (U r). Note also that, by Proposition 3.1, any left finitely presented U -module is a direct sum of modules U r (r ∈ U). Hence we are done.  5.10

∗-regular Sylvester rank functions

Now we consider the representations of ∗-rings in ∗-regular algebras. In [56] the following proposition was proved. Proposition 5.13 ([56]) Let R be a ∗-ring, U a ∗-regular ring and f : R → U a ∗-homomorphism. Then f : R → R(f (R), U ) is epic. By analogy with the notion of epic division R-rings, introduced by P. Cohn we propose the following definition. Let R be a ∗-ring. An epic ∗-regular R-ring is a triple (U , rk, f ), such that 1. U is ∗-regular ring; 2. rk is a faithful Sylvester matrix rank function on U ; 3. f : R → U is a ∗-homomorphism; 4. R(f (R), U ) = U . We will write simply (U , rk) or U instead of (U , rk, f ) if f or (rk, f ) are clear from the context. Observe that if U is a division algebra, there is only one possibility for rk. But in general this is not the case. We will say that two epic ∗-regular R-rings (U1 , rk1 , f1 ) and (U2 , rk2 , f2 ) are isomorphic if there exists an ∗-isomorphism α : U1 → U2 for which the following diagram Id

R −−−−→ ⏐ ⏐f .1

R ⏐ ⏐f .2

α

U1 −−−−→ U2 is commutative and rk2 (α(a)) = rk1 (a) for every a ∈ U1 . The following result, which follows from Proposition 5.9, shows that, as in the case of epic division R-rings, the values rk(f (M )), where M is a matrix over R, determine the epic ∗-regular ring (U , f, rk) uniquely up to isomorphism.

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Theorem 5.14 ([56]) Let (U1 , rk1 , f1 ) and (U2 , rk2 , f2 ) be two epic ∗-regular Rrings. Then (U1 , rk1 , f1 ) and (U2 , rk2 , f2 ) are isomorphic if and only if for every matrix M over R rk1 (f1 (M )) = rk2 (f2 (M )). A Sylvester matrix rank function rk on an a ∗-algebra R is called ∗-regular if there exists a ∗-algebra homomorphism f : R → U such that U is ∗-regular and rk ∈ Im f # . The previous theorem shows that the epic ∗-regular R-ring (R(f (R), U ), rk, f ) is completely determined by rk. We say that R(f (R), U ) is the ∗-regular R-algebra associated with rk. We denote by P∗reg (R) the space of ∗-regular rank functions on R. Proposition 5.15 ([56]) Let R be a ∗-algebra. Then P∗reg (R) is a closed convex subset of P(R). By Theorem 5.14, every element of P∗reg (R) has a canonical envelope if we require that this envelope has a compatible ∗-structure. It will be interesting to understand whether the same holds without this additional assumption and whether Question 5.11 has a positive solution in this particular case. Question 5.16 Let R be a ∗-ring and rk ∈ P∗reg (R). Is it true that the two questions in Question 5.11 have a positive answer for rk?

6 6.1

Algebraic reformulation of the strong Atiyah and L¨ uck approximation conjectures An algebraic variation of the strong Atiyah conjecture

In this subsection we formulate an algebraic variation of the strong Atiyah conjecture inspired by results of A. Knebusch, P. Linnell and T. Schick from [63]. First let us present Linnell’s reformulation of the strong Atiyah conjecture for torsion-free groups. Theorem 6.1 ([68]) Let K be a subfield of C closed under complex conjugation. Let G be a torsion-free group.Then G satisfies the strong Atiyah conjecture over K if and only if RK[G] is a division algebra. Let R be an algebra. We denote by K0 (R) the abelian group generated by the symbols [P ], where P runs over all finitely generated projective R-modules, with the relations [P1 ] + [P2 ] = [P3 ] if P1 ⊕ P2 ∼ = P3 . Every homomorphism f : R → S induces a map f # : K0 (R) → K0 (S) that sends [P ] to [S ⊗R P ]. For any finite subgroup H of a group G, the map K0 (K[H]) → K0 (RK[G] ) is injective. Therefore we will consider K0 (K[H]) as a subgroup of K0 (RK[G] ). Conjecture 6.2 (The algebraic Atiyah conjecture for G over K) Let K be a subfield of C closed under complex conjugation. Let G be a group with lcm(G) finite. Then {K0 (K[H])}H∈F (G) generate K0 (RK[G] ).

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In view of Theorem 6.1, if G is torsion-free, then the strong Atiyah conjecture and the algebraic Atiyah conjecture are equivalent, because for a von Neumann regular ring U the condition K0 (U) = [U ] is equivalent to U being a division algebra. In general, the algebraic Atiyah conjecture implies the strong Atiyah conjecture. In this survey we will consider only the strong Atiyah conjecture, but it will be interesting to check whether the algebraic Atiyah conjecture holds in the cases where we know that the strong Atiyah conjecture holds. 6.2

A structural reformulation of the general L¨ uck approximation conjecture

Let H be a countable group and let X be a set on which H acts on the left side. Assume that H acts freely on X and H\X is finite. We denote by UH (l2 (X)) the algebra of unbounded operators on l2 (X) commuting with the left H-action. ¯ = {x1 , . . . , xn } in X (n = |H\X|), we If we fix a set of H-representatives X obtain a natural isomorphism of H-Hilbert modules l2 (H)n and l2 (X): (a1 , . . . , an ) → a1 x1 + · · · + an xn (a1 , . . . , an ∈ l2 (H)), which induces a ∗-isomorphism ΨX¯ : UH (l2 (X)) → Matn (U (H)). Let K be a subfield of C closed under complex conjugation. Let F be a finitely generated free group. If F acts on X on the right and this action commutes with the H-action, we obtain a ∗-homomorphism fX : K[F ] → UH (l2 (X)). Now, let us use the notation of Conjecture 2.6. Fix a set of Hk -representatives ¯ k | and let ¯ k in Xk , put nk = |X X fk = ΨX¯ k ◦ fXk : C[F ] → Matnk (C[Hk ]). Remark 6.3 Note that if A ∈ K[F ], then fk (A) ∈ Matnk (K[Hk ]). Thus, the ∗-regular closure R(fk (K[F ]), Matnk (U (Hk )) of fk (K[F ]) in Matnk (U (Hk )) is contained in Matnk (RK[Hk ] ). Conjecture 2.6 claims that lim rkXk = rkG as Sylvester matrix rank functions k→∞

on K[F ]. However, observe that in general we do not know whether lim rkXk k→∞

exists. In order to avoid this difficulty we will work with rkω = lim rkXk inω stead of lim rkXk , where ω is a non-principal ultrafilter on N. Note that equality k→∞

lim rkXk = rkG is equivalent to the equality rkω = rkG for every non-principal

k→∞

ultrafilter ω on N. Therefore, we fix a non-principal ultrafilter ω on N. We can define  fω : C[F ] → Matnk (U (Hk )) ω

by sending A ∈ C[F ] to fω (A) = (fk (A)).

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Then, since {Xk } approximates G = F/N , ker fω is the ideal of C[F ] generated by {g − 1 : g ∈ N }. In particular, fω (K[F ]) ∼ = K[G]. We put  RK[G],ω = R(fω (K[F ]), Matnk (U (Hk ))). ω

Thus, RK[G],ω is a ∗-regular algebra associated with rkω ∈ P∗reg (K[G]). Now, we reformulate the general L¨ uck approximation conjecture using Theorem 5.14. In the case where G is amenable this result was proven by G. Elek in [32] and in this general form it appears in [56]. Theorem 6.4 ([56]) Let K be a subfield of C closed under complex conjugation, F a finitely generated free group and N a normal subgroup of F . For each natural number k, let Xk be an (Hk , F )-set such that Hk is a group that acts freely on Xk with finitely many orbits. Assume that the family {Xk } approximates G = F/N . Then the following two conditions are equivalent: 1. For any matrix A over K[F ], lim rkXk (A) = rkG (A).

k→∞

2. For every non-principal ultrafilter ω on N, (RK[G] , rkG ) and (RK[G],ω , rkω ) are isomorphic as epic ∗-regular K[F ]-rings.

7

The solution of the sofic L¨ uck approximation conjecture for amenable groups over fields of arbitrary characteristic

In this section we explain the proof of the following theorem. Theorem 7.1 Let K be a field and F a finitely generated free group. Let {Xk }k∈N be a family of finite F -sets. Assume that {Xk } approximates an amenable group G = F/N . Then (1) for every A ∈ Matn×m (K[F ]), there exists the limit lim rkXk (A); k→∞

(2) the limit does not depend on the sofic approximation {Xk } of G. Moreover, if we put rkG = lim rkXk ∈ Preg (K[G]) k→∞

(in view of Theorem 10.1 this is coherent with the previous definition of rkG when K is a subfied of C) and denote by dimG the associated Sylvester module rank function, then dimG is exact. Observe that the most interesting case of Theorem 7.1 corresponds to the case where K is of positive characteristic, because in the case of characteristic 0 we will prove a much stronger result in Theorem 10.1. In this general form the theorem is stated for the first time. Several particular cases were considered previously in the literature.

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1. When K = Q, in order to obtain the conclusions of the theorem, one can use the argument from [76]. A variation of this case appears also in [25]. 2. Observe that Conjecture 2.3 for amenable groups is a direct consequence of Theorem 7.1. In [31] G. Elek proved Conjecture 2.3 for amenable groups. D. Pape gave an alternative proof of this case in [95]. 3. In [30] the theorem is proved, by G. Elek, in the case when Xk are built from a Følner family. G. Elek also showed that the Sylvester module rank function dimG associated with rkG is exact. 4. In [1], it is proved a particular case of the theorem corresponding to the situation described in the parts (1) and (2) of Conjecture 1.2. This case is also considered in [14]. 7.1

Sofic approximations of amenable groups

The main idea behind the proof of Theorem 7.1 is to show that any two sofic approximations of a given amenable group are very similar. This was proved by G. Elek and E. Szab´o in [37]. Let us formulate their result. Let X be a finite set. The Hamming distance on Sym(X) is defined as follows. dH (σ, τ ) =

|{x ∈ X : σ(x) = τ (x)}| . |X|

Assume now that F is a finitely generated free group and let {Xi } be a sofic approximation of G  = F/N . Fix a non-principal ultrafilter on N and let dω be the pseudo-distance on i Sym(Xi ): dω ((σi ), (τi )) = lim dH (σi , τi ). ω



 We put Nω = {σ ∈ i Sym(Xi ) : dω (σ, 1) = 0} and Σω = i Sym(Xi )/Nω . The actions of F on Xi induce a homomorphism ψ{Xi },ω : F → Σω . Clearly ker ψ{Xi },ω = N . Now, let {Xi1 } and {Xi2 } be two sofic approximations of G = F/N . We put 1 Yi = Yi2 = Xi1 × Xi2 and let F act on Yi1 by acting only on the first coordinate and F act on Yi2 by acting only on the second coordinate. Then {Yi1 } and {Yi2 } are two approximations of F/N . Theorem 7.2 ([37, Theorem 2]) The representations ψ{Y 1 },ω and ψ{Y 2 },ω are i i conjugate. The proof of this theorem uses in an essential way the results of a fundamental work of D. Ornstein and B. Weiss [91] on amenable groups. 7.2

Proof of Theorem 7.1

Observe that an infinite subfamily of a family that approximates a group G also approximates G. Thus, if (1) or (2) does not hold we will be able to find two

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families {Xi1 }i∈N and {Xi2 }i∈N such that the limits lim rkX 1 (A) and lim rkX 2 (A) i

i→∞

i→∞

i

exist but they are different. Let us use the notation of Theorem 7.2. Then clearly rkX 1 = rkY 1 and rkX 2 = rkY 2 . i

i

i

i

On the other hand, Theorem 7.2 implies that lim rkY 1 (A) = lim rkY 2 (A) ω

i

ω

i

for any non-principal ultrafilter ω on N. Thus, lim rkX 1 (A) = lim rkY 1 (A) = lim rkY 2 (A) = lim rkX 2 (A).

i→∞

i

ω

i

ω

i

i→∞

i

We have obtained a contradiction.

8

Natural extensions of Sylvester rank functions

Let R ≤ S be two algebras and let rk ∈ P(R). In this section we consider the following question. Question 8.1 When is it possible to extend rk to a Sylvester matrix rank function on S? If there are several extensions, can we define a canonical one? We will see that if S is an “amenable” extension of R (we do not have a precise definition for this notion), then we can expect to be able to construct the “natural” extension of rk. It will be interesting to investigate further the examples presented in this section and produce a general definition for natural extensions. 8.1

A generalization of the construction of rkG

The construction of rkG may be generalized in the following may. Let S =+ R ∗ G be a crossed product of an algebra R and an amenable group G, that is S = g∈G Sg is a G-graded ring such that Se = R and for every g ∈ G there exists an invertible g¯ ∈ Sg . Let dim be an exact Sylvester module rank function on R, satisfying dim L = dim g¯L, for every g ∈ G, L ∈ R − mod.

(6)

H on S, which we will Then we can construct a Sylvester module rank function dim call the natural extension of dim. Theorem 8.2 ([112]) Let S = R ∗ G be a crossed product of an algebra R and an amenable group G and let dim be an exact Sylvester module rank function on R satisfying (6). Let M be an S-module. Then 1. Let {Fi } be a Følner family of G. For any finitely generated R-submodule K of M , there exists  dim g∈Fi g¯K e(K) = lim . i→∞ |Fi |

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2. e(K) does not depend on the Følner family {Fi }. H M = sup e(K) is an exact Sylvester module rank function on S. 3. dim K ( is associated with dim H and rk is associated with dim, we also say that rk ( is If rk ( on R is the natural extension of rk (notice that, by [112], the restriction of rk indeed equal to rk). The compatibility condition (6) can be also expressed in terms of rk: rk(A) = rk(¯ g −1 A¯ g ), for every g ∈ G and every matrix A over R.

(7)

( in terms of rk. We can also express rk Proposition 8.3 Let S = R ∗ G be a crossed product of an algebra R and an amenable group G and let rk be an exact Sylvester matrix rank function on R satisfying (7). Fix {Fi } a Følner family of G. Let A ∈ Matn×m (S) be a matrix over  S and let T be a finite set of elements of G such that the entries of A lie in g∈T Sg . Denote by φi :

1 g∈Fi

n Sg

→

1

m Sg

g∈Fi T

the R-homomorphim of free R-modules induced by right multiplication by A. Then rk(φi ) ( . rk(A) = lim i→∞ |Fi | Thus, in view of Theorem 7.1, if G is amenable, then rkG ∈ P(K[G]) is the natural extension of rkK ∈ P(K). It seems logical to ask the following questions. Question 8.4 Let S = R ∗ G be a crossed product of an algebra R and a group G and let rk ∈ P(R) be G-invariant. Is it possible to extend rk on S? Assuming that rk is faithful, is it possible to find a faithful extension? One of the motivations for these questions is Kaplansky’s direct finiteness conjecture (see Subsection 13.4). 8.2

Other instances of natural extensions

There are other instances where we can speak about the notion of natural extension. They appeared in the proof of some results from [56]. We call them algebraic and transcendental natural extensions. Let R be an algebra and rk a Sylvester matrix rank function on R. Let E/K be an algebraic extension of fields. Take a matrix A ∈ Matn×m (R ⊗K E). Then there exists a finite subextension E0 /K of E/K such that A ∈ Matn×m (R ⊗K E0 ). The action of A ∈ Matn×m (R ⊗K E0 ) by right multiplication on (R ⊗K E0 )n defines an R-homomorphism φA : (R ⊗K E0 )n → (R ⊗K E0 )m

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of free R-modules. We put rk(φA ) ( rk(A) = . |E0 : K| ( is a ( Observe that rk(A) does not depend on the choice of E0 . It is clear that rk Sylvester matrix rank function on R ⊗K E and we call it the natural algebraic extension of rk on R ⊗K E . Now consider a matrix A ∈ Matn×m (R[t]) over the polynomial ring R[t] and let i n i m φA R[t]/(ti ) : (R[t]/(t )) → (R[t]/(t )) ,

(a1 , . . . , an ) → (a1 , . . . , an )A.

We put ( i (A) = rk

) rk(φA R[t]/(ti ) i

.

Proposition 8.5 ([112, 56]) Let rk be a regular Sylvester matrix rank function. ( ( i (A), which we denote by rk(A). Then for every matrix A there exists lim rk i→∞

( Note that rk(p) = 1 for every 0 = p ∈ K[t]. Thus, taking into account a remark ( as a Sylvester rank function on R ⊗K after Proposition 5.2, we can think about rk ( on R ⊗K K(t) is called the natural K(t). The Sylvester matrix rank function rk transcendental extension of rk. As we will see later the notions of natural algebraic and transcendental extension appear in the proof of Theorem 10.1. We will use them to prove the equality between some Sylvester matrix rank functions. We can recognize the natural transcendental extension using the following result. Proposition 8.6 ([56]) Let U be a von Neumann regular algebra and let rk be a Sylvester matrix rank function on U . Let rk be a Sylvester matrix rank function on U [t±1 ] which extends rk. Assume that for any n by n matrix A, rk (In + tA) = n. Then rk is the natural transcendental extension of rk. We want to mention an interesting question, which arose when we were working on [56]. By Proposition 5.5, if U is a simple von Neumann regular ring and rk is a Sylvester matrix rank function on U such that U is complete with respect to the rk-metric, then P(U ) = {rk}. Thus, one can expect to be able to describe P(U [t]). In particular, we want to ask the following question. Question 8.7 Let U be a simple von Neumann regular ring and rk a Sylvester matrix rank function on U such that U is complete with respect to rk-metric. Let ( K = Z(U ). Is it true that P(U ⊗K K(t)) = {rk}? In [57] we answer this question positively in the case where U is a simple Artinian ring. We finish this subsection with the following general question.

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Question 8.8 Let R be an algebra and rk ∈ P(R). Let E/K a field extension. Is ( ∈ P(R ⊗K E) that unifies there a general definition for the natural extension rk the notions of algebraic and transcendental natural extensions introduced in this subsection?

9

The solution of the strong Atiyah conjecture for elementary amenable groups over fields of arbitrary characteristic

9.1

A variation of Moody’s induction theorem

Let R be an algebra. We denote by G0 (R) the abelian group generated by the symbols [M ], where M runs over all finitely generated R-modules, with the relations [M1 ] + [M3 ] = [M2 ] if there exists an exact sequence 0 → M1 → M2 → M3 → 0. If dim is an exact Sylvester module rank function on a Noetherian ring R then dim can be extended to an homomorphism dim : G0 (R) → R. Conversely any homomorphism φ : G0 (R) → R such that φ([R]) = 1 can be viewed as an exact Sylvester module rank function on R. Thus, the study of exact Sylvester module rank functions on R and of the group G0 (R) are very related. Clearly there exists a natural map K0 (R) → G0 (R). This is an isomorphism if any finitely generated R-module has a finite resolution consisting of finitely generated projective R-modules. Any flat homomorphism f : R → S induces the natural induction map f : G0 (R) → G0 (S) that sends [M ] to [S ⊗R M ]. Recall that the embedding of an algebra in an Ore ring of fractions is flat. If R ∗ G is a crossed product and H is a subgroup of G, then the embedding of R ∗ H into R ∗ G is also flat. In [86] J. Moody proved the following result. Theorem 9.1 Let R be a right Noetherian ring, let G be a polycyclic-by-finite group, and let F(G) denote the set of finite subgroups of G. Then the natural induction map 1 G0 (R ∗ H) → G0 (R ∗ G) H∈F (G)

is surjective. Corollary 9.2 ([65]) Let R be a left Artinian ring, let G be an elementary amenable group such that the orders of finite subgroups of G are bounded. Then the following holds. 1. R ∗ G satisfies the left Ore condition and the ring Ql (R ∗ G) is left Artinian. 2. The natural induction map 1 H∈F (G)

is surjective.

G0 (R ∗ H) → G0 (Ql (R ∗ G))

Jaikin-Zapirain: L2 -Betti numbers

375

Proof The first part of the corollary is proved in [65, Proposition 4.2]. Let us prove the second one. We follow the proof of [65, Lemma 4.1]. First recall an alternative description for the class of elementary amenable groups given in [65]. Let B denote the class of all finitely generated abelian- by-finite groups. For any class of groups C we denote by LC the class of locally-C groups. For each ordinal α, define Eα inductively as follows: = (LEα−1 )B if α is a successor ordinal. 0 E0 consists of trivial groups. Eα 0 Eα = β f dμ = f (T )v, v , for all continuous functions f on Spec(T ). Spec(T )

In particular, we have μ(Spec(T )) = ,v,2 ; so it is a finite measure. The measure μ from the proposition is called the spectral measure associated to v and T . In a similar way we can associate to the operators φA X probability C ∗ )|A| (see Lemma 10.4). It Radon measures μA on [0, a], where a = S(A)S(A X can be done in the following way.

Jaikin-Zapirain: L2 -Betti numbers

380

¯ of representatives of H-action on X. For each x ¯ and 1 ≤ i ≤ n, Fix a set X ¯∈X A let (μX )x¯,i be the Radon measure associated to (0, . . . , x ¯, . . . , 0) (¯ x is on the ith place) and φA X . Now, we put 1 μA X = ¯ |X|



(μA ¯,i . X )x

¯ x ¯∈X,1≤i≤n

A If G is a group, then μA G will denote the measure associated with φG . Let S be a metric space with its Borel σ-algebra Σ. We say that a sequence of positive probability measures μi (i ∈ N) on (S, Σ) converges weakly to the measure μ, if > > S

f dμi →

S

f dμ (when i → ∞)

for all bounded, continuous functions f on S. From now on, let F be a finitely generated free group and N a normal subgroup of F . For each natural k, let Xk be an (Hk , F )-set such that Hk is a countable group that acts freely on Xk and Hk \Xk is finite. Assume that {Xk } approximates G = F/N . Let A = BB ∗ for some B ∈ Matn×m (K[F ]). A Lemma 10.6 The measures μA Xk converge weakly to μG .

Proof We should check that for any continuous function f on [0, a] > > f dμA → f dμA Xk G. [0,a]

[0,a]

Since, by the Weierstrass Approximation Theorem, any continuous function can be approximated by polynomials, we can assume that f = xi . Note that > [0,a]

xi dμA Xk =

i TrHk (φA Xk )

|Hk \Xk |

i

=

TrHk φA Xk |Hk \Xk |

.

Now, since Xk approximate G, we obtain that i

TrHk φA Xk |Hk \Xk |

k→∞

i

>

−−−→ TrG φA G =

xi dμA G. [0,a]

 Clearly the previous lemma does not imply directly that μA Xk ({0}) converges to ({0}) (note that this is an equivalent reformulation of Conjecture 2.6). However, μA G it implies one of the two inequalities of Conjecture 2.6. Proposition 10.7 (Kazhdan’s inequality) The following inequality holds: A lim sup dimXk ker φA Xk ≤ dimG ker φG . k→∞

Jaikin-Zapirain: L2 -Betti numbers

381

Proof Note that by the Portmanteau theorem (see, for example, [27, Theorem 11.1.1]), A μA (8) G (C) ≥ lim sup μXk (C) for all closed sets C of [0, a]. k→∞

Thus, we obtain the following A A A dimG ker φA G = μG ({0}) ≥ lim sup μXk ({0}) = lim sup dimXk ker φXk . k→∞

k→∞

 10.3

The determinant conjecture

Observe that the Portmanteau theorem implies also that for  > 0, A A μA G ({0}) ≤ μG ([0, )) ≤ lim inf μXk ([0, ))

≤ lim inf k→∞

k→∞ A μXk ({0}) +

lim sup μA Xk ((0, )), k→∞

and so A A μA G ({0}) − lim inf μXk ({0}) ≤ lim sup μXk ((0, )) k→∞

k→∞

Thus, in order to prove Conjecture 2.6, it will be enough to show that μA Xk ((0, )) tends uniformly (in k) to zero when  tends to zero. With this aim, it was proposed to use the Fuglede-Kadison determinant of φA X defined as follows. :J ;  a exp 0+ ln(x)dμA if the integral converges + A X det (φX ) := 0 otherwise This idea is contained implicitly in the paper of W. L¨ uck [76]. It seems that explicitly it appeared first in [25, 104]. Proposition 10.8 Assume that there exists a constant C such that ln det+ (φA Xk ) ≥ C for all k. Then A μA G ({0}) = lim μXk ({0}). k→∞

Moreover

ln det+ (φA G)

≥ C as well. >

Proof Assume that

a 0+

ln(x)dμA Xk ≥ C.

Hence, for any  > 0, > μA Xk ((0, )) ln  ≥

0+

> ln(x)dμA Xk ≥ C −

a

ln(x)dμA Xk ≥ C − a ln a.

Jaikin-Zapirain: L2 -Betti numbers

382 Thus, we obtain that μA Xk ((0, )) ≤

a ln a − C . − ln 

Thus, μA Xk ((0, )) tends uniformly (in k) to zero and so A μA G ({0}) = lim μXk ({0}). k→∞

This proves the first statement of the proposition. The second statement follows from the Portmanteau theorem. > a > a A ln det+ (φA ) = ln(x)dμ = lim ln(x)dμA G G G

→0+

0+

>

by (8)

≥

a

lim lim sup

→0+

k→∞

ln(x)dμA Xk ≥ C. 

Thus, the previous proposition shows that Conjecture 2.6 is a consequence of the following conjecture. Conjecture 10.9 (The determinant conjecture over K) Let K be a subfield of C closed under complex conjugation. Let F be a finitely generated free group and A a ∗-symmetric matrix over K[F ]. Then there exists a constant C depending only on A such that for every countable group H and every (H, F )-set X such that H acts freely on X and H\X is finite, ln det+ (φA X ) ≥ C. As we have seen before the determinant conjecture is a way to control the measures μA X uniformly in the small intervals around 0. A stronger form of the determinant conjecture is a conjecture of J. Lott and W. L¨ uck (formulated only when K = Q) about Shubin-Novikov invariants (see [78]). It was known for free groups ([101]) and free abelian groups ([73, 80]) but few years ago a counterexample was constructed by L

. Grabowski [43]. Another related conjecture is the determinant approximation conjecture. We will not describe it here, but we recommend to read the introduction of [43], where the relation between all three conjectures is presented. Unfortunately Conjecture 10.9 is not correct if K = C. Our example is a modification of [78, Example 13.69]. ∞ 1 Construct a sequence n1 = 1 and nj+1 = 3nj . Put r = j=1 nj . Consider A = z − e2πir ∈ C[z ±1 ] = C[Z]. Let Xj = Z/(nj ). Then ln det+ φA Xk ≤ ln |e

2πi

∞

1 j=k+1 nj

− 1| + (nk − 1) ln 2

≤ − ln nk+1 + nk ln 2 = (ln 2 − ln 3)nk . Thus, ln det+ φA Xk are not bounded from below.

Jaikin-Zapirain: L2 -Betti numbers 10.4

383

¯ The proof of the determinant conjecture for sofic groups over Q

The following theorem is a slight modification of [26, Theorem 3.2]. Theorem 10.10 Let X be an (H, F )-set such that F is a finitely generated free group and H is a group acting freely on X with finite number of orbits. Let K be a ¯ the s different embeddings number field of degree s over Q. Denote by σi : K → Q ∗ with σ1 = Id. Let A = BB with B ∈ Matn×m (OK [F ]) (OK is the ring of integers of K). If H is sofic, then ln det+ (φA X ) ≥ −n

s 

ln(

C

S(σi (A))S(σi (A)∗ )|A|).

i=2

+ Proof First we consider the case when H is trivial (or finite). Let C = si=1 σi (A) be the diagonal sum of the matrices σi (A). Note that C is not a ∗-symmetric pos|X| as the product of the absolute itive matrix, but we still can define (det+ φC X) values of all non-zero roots (counted with multiplicities) of the characteristic poly+ σi (A) )|X| as the product of the absolute values of all nomial of φC X and (det φX non-zero roots (counted again with multiplicities) of the characteristic polynomial σ (A) |X| is a non-zero algebraic integer lying in Q, and of φXi . In particular (det+ φC X) + C |X| so, (det φX ) ≥ 1. Hence 1 ≤ det+ φC X =

s 

σ (A)

det+ φXi

s 

≤ det+ φA X

i=1

σ (A) n

,φXi

,

i=2 Lemma 10.4

≤

det+ φA X

s C  ( S(σi (A))S(σi (A)∗ )|A|)n i=2

Therefore, we conclude that ln det+ φA X ≥ −n

s 

ln(

C

S(σi (A))S(σi (A)∗ )|A|).

i=2

Now, assume that H is sofic. As we have explained in Subsection 10.1, we can AX ¯ ˜ represent the operator φA X in the form φH ˜ . Let {Yk } be a family of finite F -sets ˜ that approximate H. Then from the finite case of the proposition we obtain that A

ln det+ φYkX¯ ≥ −n

s 

ln(

C

S(σi (AX¯ ))S(σi (AX¯ )∗ )|AX¯ |)

i=2 by Lemma 10.3

≥

−n

s 

ln(

C

S(σi (A))S(σi (A)∗ )|A|).

i=2

Thus, by Proposition 10.8, ˜ A

¯ + X ln det+ φA X = ln det φH ˜ ≥ −n

s 

ln(

C S(σi (A))S(σi (A)∗ )|A|).

i=2



Jaikin-Zapirain: L2 -Betti numbers

384

¯ Theorem 10.10 and Proposition 10.8 imply together Theorem 10.1 with K = Q. ¯ ]) Corollary 10.11 Let F be a free finitely generated group, A ∈ Matn×m (Q[F and Hk (k ∈ N) a family of sofic groups. For each natural number k, let Xk be an (Hk , F )-set such that Hk acts freely on Xk and Hk \Xk is finite. Assume that {Xk } approximates G = F/N . Then A lim dimXk ker φA Xk = dimG ker φG .

k→∞

Proof Without loss of generality we may assume that A = BB ∗ where B is a matrix over OK [F ] and K is a finite extension of Q. Now we can apply Theorem 10.10 and Proposition 10.8.  10.5

The proof of Theorem 10.1

¯ Now let In the previous section we have proved Theorem 10.1 in the case K = Q. us explain how to proceed in the general case. This has been done recently in [56]. Assume, in addition, that K is closed under complex conjugation. Then rkG is a ∗-regular Sylvester matrix rank function on K[G]. The ∗-regular algebra associated to rkG is RK[G] . Let us fix a non-principal ultrafilter ω on N. Then rkω = lim rkXk ω

is another ∗-regular Sylvester matrix rank function on K[G]. The ∗-regular K[G]algebra associated with rkω is RK[G],ω . A straightforward reformulation of Theorem 10.1 is to say that for every nonprincipal ultrafilter ω on N, rkG = rkω as Sylvester matrix rank functions on K[G]. Our structural reformulation of the sofic L¨ uck approximation conjecture over K (Theorem 6.4) implies that it is equivalent to the existence of a K[G]-*-isomorphism αK : RK[G] → RK[G],ω such that rkG = rkω ◦αK . At first glance, it seems that this reformulation cannot help us to prove Theorem 10.1, because to prove the existence of αK is harder than to prove the equality between the Sylvester rank functions rkG and rkω . However, we have already ¯ Thus, we know proved Theorem 10.1 when K = Q (and in fact, when K = Q). that αQ exists! This is the first brick in our construction of αK for an arbitrary subfield K of C. It is clear that it is enough to prove Theorem 10.1 for finitely generated subfields K of C. Any finitely generated subfield K of C of transcendental degree n over Q is a subfield of a field K2n , where Ki are constructed inductively: 1. K1 = Q; 2. if i ≥ 1, K2i = K2i−1 is the algebraic closure of K2i−1 in C; 3. if i ≥ 1, K2i+1 = K2i (λi ) for some λi ∈ C \ K2i such that |λi | = 1.

Jaikin-Zapirain: L2 -Betti numbers

385

Theorem 10.1 for Ki is proved by induction on i. First we consider the inductive step for algebraic extensions. Thus, we assume that Theorem 10.1 holds for K2i−1 . Given a Sylvester matrix rank function rk on an algebra R and an algebraic extension E/K we have defined in Subsection 8.2 the natural algebraic extension ( ∈ P(R ⊗K E) of rk. It is proved in [56] that if G is sofic, K is a subfield of C rk closed under complex conjugation and the sofic L¨ uck approximation holds over K, then ∼ ¯ ¯ RK[G] as K[G]-∗-rings (9) = RK[G] ⊗K K ¯ is the natural algebraic extension of and, moreover, the restriction of rkG on RK[G] ¯ the restriction of rkG on RK[G] . This also implies a similar statement for rkω : the restriction of rkω on RK[G],ω is the natural algebraic extension of the restriction of ¯ rkG on RK[G],ω and ∼ ¯ RK[G],ω = RK[G],ω ⊗K K ¯

¯ as K[G]-∗-rings.

(10)

Using the induction assumption we have that there exists αK2i−1 . Taking into account (9) and (10), we construct αK2i . Now the uniqueness of the natural extension implies that rkω ◦αK2i = rkG as Sylvester matrix rank functions on K2i [G]. This proves Theorem 10.1 for K2i . Let us describe now the proof of the inductive step for transcendental extensions. We assume that Theorem 10.1 holds for K2i . Given a regular Sylvester matrix rank function rk on an algebra R we have ( ∈ P(R ⊗K K(t)) defined in Subsection 8.2 the natural transcendental extension rk ( is of rk. If R is a von Neumann regular algebra then, by Proposition 8.6, rk characterized by the condition that for every n by n matrix A over R, ( n + tA) = n. rk(I This leads us to consider the following conjecture. Conjecture 10.12 (The strong algebraic eigenvalue conjecture over K for G) Let G be a countable group, K a subfield of C and A ∈ Matn (RK[G] ). Then for any λ ∈ C which is not algebraic over K, the matrix A − λIn is invertible over U (G). This conjecture generalizes the algebraic eigenvalue conjecture formulated in [26]. The proof of the strong algebraic eigenvalue conjecture for a sofic group G over an arbitrary subfield of C is presented in [56]. Observe that the strong algebraic eigenvalue conjecture over K2i implies that rkG (In + λi A) = rkG (A + λ−1 i In ) = n for every n by n matrix A over RK2i [G] . This means that the restriction of rkG on K2i+1 [G] is the natural transcendental extension of the restriction of rkG on K2i [G]. The same holds for rkω . The uniqueness of the natural transcendental extension implies that rkG and rkω as Sylvester matrix rank functions on K2i+1 [G] are equal. This proves Theorem 10.1 for K2i+1 .

Jaikin-Zapirain: L2 -Betti numbers

386 10.6

Other variations of the L¨ uck approximation

There are other variations of the L¨ uck approximation considered in the literature. The L¨ uck approximation in the context of Benjamini-Schramm convergence of graphs is studied by M. Ab´ert, A. Thom and B. Virag in [3]. In [61] S. Kionke describes a general construction of the Sylvester matrix rank function on C[G] associated with a representation of the group G in a finite von Neumann algebra and extends the L¨ uck approximation to this more general situation. In [96] H. D. Petersen, R. Sauer and A. Thom present a general L¨ uck approximation theorem for normalised Betti numbers for Farber sequences of lattices in totally disconnected groups.

11

The Approximation and strong Atiyah conjecture for completed group algebras of virtually pro-p groups

Let (R, m) be a commutative completed local domain such that R/m is finite of characteristic p > 0 and let K be the ring of fractions of R. In this section K may be of characteristic p or 0. Let G be a countably based virtually a pro-p group. We consider Λ = Λ(R[[G]]) = K ⊗R R[[G]] (see [23] for the definition of R[[G]] and its properties). For every open normal subgroup U of G we have the canonical map Λ → K[G/U ], which induces a Sylvester matrix rank function rkG/U on Λ. Let us formulate the L¨ uck approximation and the strong Atiyah conjectures in this situation. Conjecture 11.1 Let G be virtually a pro-p group and let G > G1 > G2 > . . . be a chain of open normal subgroups of G with trivial intersection. Then 1. there exists the limit lim rkG/Gi ∈ P(Λ); i→∞

2. the limit does not depend on the chain G > G1 > G2 > . . . ; 1 3. if lcm(G) < ∞, lim rkG/Gi (A) ∈ Z for every matrix A over Λ. i→∞ lcm(G) If the parts (1) and (2) of conjecture hold we denote the limit by rkR[[G]] . Then dimR[[G]] will denote the Sylvester module rank function associated with rkR[[G]] . We can also give an explicit formula for dimR[[G]] . If M is finitely presented Λmodule, we obtain dimK K[G/Gi ] ⊗Λ M . i→∞ |G : Gi |

dimR[[G]] M = lim

(11)

Observe that Conjecture 11.1 is stronger than Conjecture 1.2, because the first conjecture claims the approximation for matrices over Λ, which is larger than K[G]. However since G is virtually pro-p, the case of Conjecture 11.1 (1) and (2), where K is of characteristic p, is easier than the one where K is of characteristic 0.

Jaikin-Zapirain: L2 -Betti numbers

387

Proposition 11.2 ([14, Lemma 4.1]) Assume that K is of characteristic p. Then the parts (1) and (2) of Conjecture 11.1 hold. Proof When Gi+1 is pro-p, K[Gi /Gi+1 ] is a local ring. Thus, for this i, we obtain that for every finitely presented Λ-module M , dimK (K[G/Gi ] ⊗Λ M ) |G : Gi | dimK (K[G/Gi ] ⊗K[G/Gi+1 ] (K[G/Gi+1 ] ⊗Λ M ) ) = |G : Gi | dimK (K[G/Gi+1 ] ⊗Λ M ) ≥ |G : Gi+1 | = dimG/Gi+1 M.

dimG/Gi M =

This shows that the limit (11) exists. A similar argument shows that it does not depend on the chain (see [55, Corollary 2.2]).  Later we will use the following property of dimR[[G]] when R is of characteristic p. Proposition 11.3 Assume that K has characteristic p. Let M be a proper quotient of Λn . Then dimR[[G]] M < n. Proof Since M is a proper quotient of Λn , there exists l such that if i ≥ l, dimG/Gi M < n. As we have seen in the proof of the previous proposition {dimG/Gi M } is virtually decreasing. Hence dimR[[G]] M < n  Recall that a p-adic analytic profinite group can be defined as a closed subgroup of GLn (Zp ) for some n. These groups play a special role in the theory of pro-p groups and linear groups (see [23]). For example, A. Lubotzky [75] proved the following criterion of linearity of a finitely generated group. Proposition 11.4 ([75]) Let G be a finitely generated group. Then G is linear over a field of characteristic 0 if and only if G can be embedded into a p-adic analytic group for some prime p. If G is a p-adic profinite group and R is Noetherian, then R[[G]] is a Noetherian ring ([23]). By a result of Lazard [66], if G is torsion free, then the ring R[[G]] does not have non-trivial zero divisors (see also [87, 8, 108]). Thus, in this case the ring DR (G) = Ql (R[[G]]) is a division algebra. An arbitrary p-adic analytic profinite groups G contains an open torsion-free normal pro-p subgroup H. Therefore, we can speak about rkDR (H) ∈ P(R[[H]]). Observe that R[[G]] = R[[H]]∗G/H. So, we define rkDR (G) as the natural extension of rkDR (H) (see Subsection 8.1). The following result is attributed to M. Harris [52] (see also [39, 14]).

Jaikin-Zapirain: L2 -Betti numbers

388

Proposition 11.5 ([52]) Let G be a p-adic analytic profinite group. Then rkR[[G]] = rkDR (G) . In particular, 1. Conjecture 11.1 (1) and (2) hold for p-adic analytic profinite groups and 2. Conjecture 11.1 (3) holds for torsion-free p-adic analytic profinite groups. As a consequence we obtain that Conjecture 11.1 (3) holds over fields of characteristic p for a large class of pro-p groups which includes free pro-p groups. Corollary 11.6 ([55]) Let K be of characteristic p. Let G be an inverse limit of torsion-free p-adic pro-p groups. Then Conjecture 11.1 (3) holds for G. Proof We combine the approximation in characteristic p (Proposition 11.2) and Proposition 11.5.  At this moment Corollary 11.6 describes all the examples for which Conjecture 11.1 (3) is known when K has characteristic p. It will be interesting to prove Conjecture 11.1 (3) over fields of characteristic p for virtually free pro-p groups and free-by-cyclic pro-p groups. When K has characteristic 0, the obvious example to check is the case of free pro-p group. All three parts of Conjecture 11.1 are not known in this case.

12 12.1

Positive results on the strong Atiyah conjecture over fields of characteristic 0 Amenable extensions

Let F be a free group freely generated by a finite set S and N a normal subgroup of F . Put G = F/N . Let H be a normal subgroup of G such that G/H is amenable. ¯ of H in G. Since G/H is amenable we can find a family Consider a transversal X ¯ ¯ satisfying of finite subsets {Xk }k∈N of X   1 ¯ k |, where T¯k = {¯ ¯k : H x ¯ k }. |X x∈X ¯(S ∪ S −1 ∪ {1})k+1 ⊆ H X |T¯k | ≥ 1 − k (12) ¯ k and Tk = H T¯k . Our aim is to define a right action of F on We put Xk = H X Xk that commutes with the left action of H and such that the (H, F )-sets Xk approximate G. ¯ k we will define an element x First for any s ∈ S and any x ¯ ∈ X ¯ · s ∈ Xk . If −1 k x ¯ ∈ Tk (1 ∪ S ∪ S ) , then x ¯·s = x ¯s ∈ Xk is well defined by our conditions. If x ¯ ∈ Tk (1 ∪ S ∪ S −1 )k we define x ¯ · s ∈ Xk in a such way that the induced action of ¯ k is a bijection. s on H\Xk = H\H X Now, if x ∈ Xk is an arbitrary element we can write x = h¯ x for some h ∈ H and ¯ k and we define x · s = h(¯ x ¯∈X x · s). Thus, Xk is an (H, F )-set, H acts freely on Xk and H\Xk is finite.

Jaikin-Zapirain: L2 -Betti numbers

389

Let x ∈ Tk and w be a word in S of length l ≤ k. Arguing by induction on l, we easily obtain that x · w = xw. Hence we obtain that Xk approximate G. In the following theorem we show that the general L¨ uck approximation over C holds in the previous situation. Our proof is similar to the one of [76, Theorem 6.37]. Theorem 12.1 Let A ∈ Matn×m (C[F ]). Then A lim dimXk ker φA Xk = dimG ker φG .

k→∞

Proof For simplicity we assume that A ∈ C[F ]. The general case can be proved similarly. If Z is a subset of G we denote by Z c = G \ Z the complement of Z in G. For ¯ we denote by projY¯¯2 the projection of l2 (H Y¯2 ) subsets Y¯ , Y¯1 ⊂ Y¯2 subsets of X Y1 onto l2 (H Y¯1 ). 2 c G 2 For every k ≥ 1 we put Uk = ker φG A ∩ l ((Xk ) ), Wk = ker φA ∩ l (Tk ) and Lk = (Uk ⊕ Wk )⊥ ∩ ker φG . Then U , W and L are (left) H-invariant closed k k k A subspaces and the following decomposition holds ker φG A = U k ⊕ Wk ⊕ L k . Let k0 ≥ 1 be such that all the group elements involved in A lie in S k0 and let k ≥ 2k0 . By the definition of the sets T¯k (see (12)), we obtain that (Xk )c · S k0 ∩ Tk · S k0 = ∅, and so, 2 c 2 ker φG A ∩ (l ((Xk ) ) ⊕ l (Tk )) = Uk ⊕ Wk . ¯

Hence the restriction of projX ¯ X

k \T k

¯

¯

on Lk is injective and so ¯

X dimH projX ¯ (Lk ) = dimH projX ¯ X k

k \T k

¯

¯ k \ T¯k |. (Lk ) ≤ |X

(13)

k The definition of the sets T¯k also implies that Wk ≤ ker φX A . We put Mk = Xk ⊥ (Wk ) ∩ ker φA . Then Mk is a (left) H-invariant closed subspace and k ker φX A = Wk ⊕ Mk .

¯

k Since the restriction of projX ¯ \T¯ on Mk is injective, we obtain that X k

k

¯ k \ T¯k |. dimH Mk ≤ |X Observe that on the one hand (1), 1 dimG ker φA G = projker φA G  1 = ¯ projker φA (g), g G | Xk | ¯k g∈X

1  projUk (g) + projWk (g) + projLk (g), g = ¯ | Xk | ¯k g∈X

¯

=

dimH Wk + dimH projX ¯ k (Lk ) X ¯ | Xk |

(14)

Jaikin-Zapirain: L2 -Betti numbers

390 and on the other

1 A dimXk ker φA Xk = ¯ dimH ker φXk | Xk | 1  = ¯ projker φA (g), g Xk | Xk | ¯ g∈Xk

1  projWk (g) + projMk (g), g = ¯ | Xk | ¯ g∈Xk

dimH Wk + dimH Mk . = ¯k | |X Thus, we have that

9 9 ¯ 9 9 (L ) − dim M 9dimH projX 9 H k k 9 9 ¯ Xk A 9 9dimG ker φA = ker φ − dim Xk G Xk ¯ | Xk | by (13) and (14)

≤

¯ k \ T¯k | 1 |X ¯k | ≤ k . |X 

This finishes the proof of the proposition.

Corollary 12.2 Let G be a countable group and let A be a normal subgroup of G such that G/A is amenable. Then rkG as a Sylvester matrix rank function on RK[A] ∗ G/A is a natural extension of the Sylvester matrix rank function rkA on RK[A] . Proof By Proposition 5.13, RK[G] is an epic ∗-regular K[G]-ring. Therefore, by Proposition 5.9, rkG as a Sylvester matrix rank function on RK[G] is completely determinated by its values on matrices over K[G]. Hence, Theorem 12.1 and Proposition 8.3 imply that rkG as a Sylvester matrix rank function on RK[A] ∗ G/A is the natural extension of rkA .  Corollary 12.3 Let G be a group such that lcm(G) < ∞ and let H be a normal subgroup of G such that G/H is elementary amenable. Let K be a subfield of C. Assume that for every finite subgroup P/H of G/H, P satisfies the strong Atiyah conjecture over K. Then G satisfies the strong Atiyah conjecture over K. Proof Let α be the least ordinal such that G/H ∈ Eα . We argue by transfinite induction on α as we have done in the proof of Corollary 9.2. We use the notation introduced there. We can assume that α = γ + 1, and there exists a normal subgroup A/H ∈ LE γ of G/H such that G/A ∈ B. Notice that if T /A is a finite subgroup of G/A, then T ∈ LE γ . Hence T satisfies the strong Atiyah conjecture over K. By Proposition 5.9, rkT (r) : r ∈ RK[T ]

by Proposition 5.9

=

AK (T ) =

1 1 Z≤ Z. lcm(T ) lcm(G)

Thus, Corollary 12.2 and Corollary 9.3 imply that G satisfies the strong Atiyah conjecture over K. 

Jaikin-Zapirain: L2 -Betti numbers 12.2

391

The strong Atiyah conjecture for groups from the class D

The class D is the smallest non-empty class of groups such that: 1. If G is torsion-free and A is elementary amenable, and we have a projection p : G → A such that p−1 (E) ∈ D for every finite subgroup E of A, then G ∈ D. 2. D is subgroup closed. 3. Let Gi ∈ D be a directed system of groups and G its (direct or inverse) limit. Then G ∈ D. Theorem 12.4 ([26, 56]) Let G be a group from the class D. Then G satisfies the strong Atiyah conjecture over C. Proof For ordinals α define the class of groups Dα as follows: 1. D0 is the class of torsion-free elementary amenable groups. 2. Dα+1 is the class of groups G such that G is a subgroup of a direct or inverse limit of groups Gi ∈ Dα or G is torsion-free and there exists an elementary amenable group T , and a map p : G → T such that p−1 (E) ∈ Dα for every finite subgroup E of T . 0 3. Dβ = α G1 > G2 > . . . be a descending  chain of subgroups such that Gi is normal in G, the index |G : Gi | is finite and i≥1 Gi = {1}. We put Xi = X/Gi . Then Xi is a normal cover of X. We may ask the following natural questions. Question 13.1 Let K = Q or Fp and let p ∈ N.



1. How do the normalized pth Betti numbers over K, bp (Xi , K) ? i→∞ |G : Gi |

 bp (Xi , K) , grow? Is |G : Gi |

there lim



2. How does the growth of the numbers the chain G > G1 > G2 > . . . ?

bp (Xi , K) |G : Gi |

 depend on the choice of

Let us show that these two questions are reformulations of the first two questions of Question 1.1. Consider the cellular chain complex of X

∂p+1

∂p

C(X) : . . . → Z[Cp+1 (X)] → Z[Cp (X)] → Z[Cp−1 (X)] → . . . → Z → 0. Since G acts freely and X/G is of finite type, we obtain that Z[Cp (X)] is a free Z[G]-module of finite rank and the connected morphisms ∂p are represented by a multiplication by a matrix over Z[G]. Hence we obtain the following representation of C(X) C(X) : . . . → Z[G]np+1

×Ap+1

×Ap

→ Z[G]np → Z[G]np−1 → . . . → Z → 0.

Therefore the normalized pth Betti numbers over K, puted as

 bp (Xi , K) , can be com|G : Gi |

bp (Xi , K) dimK Hp (Gi \C, K) = = np − (rkG/Gi (Ap ) + rkG/Gi (Ap+1 )). |G : Gi | |G : Gi | Thus, the answer to Question 13.1 is known when K = Q (Theorem 10.1) and when K = Fp and G is amenable (Theorem 7.1). Moreover, in the case K = Q the bp (Xi , K) limit lim is equal to the pth L2 -Betti number of X: i→∞ |G : Gi | 2 b(2) p (X) = dimG Hp (X, RQ[G] ) = dimG Hp (X, l (G)).

Jaikin-Zapirain: L2 -Betti numbers 13.2

395

A characterization of amenable groups G

When G is amenable we have seen in Theorem 7.1 that the Sylvester module rank function dimG on K[G] is exact. This implies the following result proved first by M. Cheeger and M. Gromov [18]. Theorem 13.2 ([18]) Let X be an aspherical CW-complex and let an amenable (2) group G acts freely on X. Then bp (X) = 0 for any p ≥ 1. In [78, Conjecture 6.8] W. L¨ uck conjectured that the property proved in the theorem characterizes amenable groups. This has been confirmed recently by L. Bartholdi [11]. In fact, his result implies the following elegant characterization of amenable groups. Theorem 13.3 ([11]) Let G be a group and let K be a field. Then G is amenable if and only if K[G] has an exact Sylvester module rank function. Proof If G is amenable then dimG is exact. If G is not amenable, then L. Bartholdi [11] proved that there exists n ∈ N such that K[G]n+1 is isomorphic to a submodule of K[G]n . This clearly implies that K[G] does not have an exact Sylvester module rank function.  13.3

The growth of the first Fp -Betti numbers of subgroups of finite index of a finitely presented group

A particular case of the situation described in Subsection 13.1 is the study of the growth of the first Betti numbers of subgroups of finite index of a finitely presented group. Conjecture 13.4 Let G be a finitely presented group and let G > G1 > G2 > . . . be a descending chain of subgroups such that Gi is normal in G, the index |G : Gi |  b1 (Gi , Fp ) exists and, moreover, is finite and i≥1 Gi = {1}. Then lim i→∞ |G : Gi | lim

i→∞

b1 (Gi , Fp ) (2) = b1 (G). |G : Gi |

(2)

The number b1 (G) is the first L2 -Betti number of G and it is defined as (2)

b1 (G) = dimG H1 (G, RQ[G] ) = dimG H1 (G, l2 (G)). Conjecture 13.4 is discussed, for example in [38]. It is related to another interesting problem. Recall that d(G) denotes the minimal number of generators of G. Conjecture 13.5 Let G be a finitely presented group and let G > G1 > G2 > . . . be a descending  chain of subgroups such that Gi is normal in G, the index |G : Gi | is finite and i≥1 Gi = {1}. Then lim

i→∞

d(Gi ) (2) = b1 (G). |G : Gi |

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We will not say much about this conjecture and recommend to look at the following paper of M. Abert and N. Nikolov [2]. A very interesting particular case of Conjecture 13.4 (proposed by F. Calegari and M. Emerton in [16]) arises when G is a lattice in SL2 (C) and {Gi } is a p-adic chain, i.e., the completion of G with respect to {Gi } is a p-adic group. The interest of F. Calegari and M. Emerton in this question was motivated by questions in the theory of automorphic forms [17]. Another motivation comes from the paper [49], where it is shown that if the Calegari-Emerton conjecture holds then the congruence kernel of any arithmetic lattice in SL2 (C) is a projective profinite group. 13.4

Kaplansky’s conjectures about group algebras

Let G be a group and K a field. I. Kaplansky proposed several conjectures about the group ring K[G]. We say that a ring R is directly finite if xy = 1 in R implies that yx = 1 as well. Kaplansky’s direct finiteness conjecture states that the group ring K[G] is directly finite (see [28] for more details about this problem). The following observation provides a large source of rings which are directly finite. Proposition 13.6 Assume that an algebra R has a faithful Sylvester matrix rank function. Then Matn (R) is directly finite for all n. Proof We will prove the statement for R, the same proof works for Matn (R). Let rk be a faithful Sylvester matrix rank function on R. Assume xy = 1. In particular, rk(x) = 1, Then, by Proposition 5.1(2), 0 = rk(x(yx − 1)) ≥ rk(yx − 1). 

Since rk is faithful, yx = 1.

Corollary 13.7 ([60, 34]) Let G be a group and K a field. Assume that K is of characteristic 0 or K is of positive characteristic and G is sofic. Then Matn (K[G]) is directly finite for all n. Proof Clearly, without loss of generality, we can assume that K and G are finitely generated. Therefore, K is a subfield of C if the characteristic of K is zero. The Sylvester matrix rank function rkG on K[G] is faithful, and so, we can apply Proposition 13.6. Assume now that K is of positive characteristic and G is sofic. Represent G as G = F/N , where F is a finitely generated free group. Let {Xk } be a family of finite F -sets approximating G. Fix a non-principal filter ω on N and let rkω = lim rkXi ∈ ω

P(K[F ]). Since rkω (g − 1) = 0 for every g ∈ N , rkω is also a Sylvester matrix rank function on K[F/N ] = K[G]. In order to apply Proposition 13.6 we have to show that rkω is faithful on K[G].  Let a = li=1 ai gi ∈ K[G] (0 = ai ∈ K, gi ∈ G) be such that all gi are different.  Consider fi ∈ F such that gi = fi N . Put A = li=1 ai fi ∈ K[F ].

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Since {Xi } approximate F/N , for any  > 0 there is n ∈ N such that for every j ≥ n there exists a subset Lj of Xj of size at least (1 − )|Xj | such that xfi = xf1 for every x ∈ Lj if i = 1. L let us construct inductively a subset {x1 , x2 , . . . , xm } of Lj , where m = K Now |Lj | l

. Let x1 be any element of Lj . Assume we have constructed x1 , . . . , xt . Then take any l  xt+1 ∈ Lj \ {x1 , . . . , xt }fi f1−1 . i=1

Then xt+1 f1 ∈

l 

{x1 , . . . , xt }fi ∪

i=1

From (15) we obtain that (

t+1 i=1

l 

xt+1 fi .

(15)

i=2

αi xt )A = 0 if αt+1 = 0. Therefore,

dimK (

m 

Kxi )A = m.

i=1

Hence, 1 rkω (a) = rkω (A) = lim rkXi (A) ≥ . ω l  Now, assume that G is torsion-free. Kaplansky’s zero divisor conjecture states that K[G] does not contain non-trivial zero divisors, that is, it is a domain. In view of Proposition 5.1(2), an algebra, having a faithful Sylvester matrix rank function taking only integer values, is a domain. Thus, the strong Atiyah conjecture over K implies Kaplansky’s zero divisor conjecture for K[G]. There are cases of groups G where we know Kaplansky’s zero divisor conjecture for C[G], but we still do not know the strong Atiyah conjecture for G. This is the case of one-relator groups without torsion. Problem 13.8 Show that the strong Atiyah conjecture holds for one-relator groups without torsion. By a result of D. Wise [114], the one-relator groups with torsion are virtually cocompact special. Hence in this case the strong Atiyah conjecture follows from Theorem 12.9. 13.5

The Hanna Neumann conjecture

Let F be a free group and U and W two finitely generated subgroups of F . In 1954, A. G. Howson [54] showed that the intersection of U and W is finitely generated. Three years later H. Neumann [88] improved the Howson bound and proved that d(U ∩ W ) ≤ 2d(U )d(W ) where d(U ) = max{d(U ) − 1, 0}.

Jaikin-Zapirain: L2 -Betti numbers

398

She also conjectured that the factor of 2 in the above inequality is not necessary and that one always has d(U ∩ W ) ≤ d(U )d(W ). This statement became known as the Hanna Neumann conjecture. It received a lot of attention since then. In 1990, W. D. Neumann [89] conjectured that, in fact, the following inequality holds  d(U ∩ xW x−1 ) ≤ d(U )d(W ). x∈U \F/W

This conjecture received the name of the strengthened Hanna Neumann conjecture. It was proved independently by J. Friedman [40] and I. Mineyev [85] in 2011. Later W. Dicks gave a simplification for both proofs (see [21, 40]). In [55] A. Jaikin-Zapirain gave a new proof of the strengthened Hanna Neumann conjecture. This new approach used the L¨ uck approximation and the strong Atiyah conjecture for free groups. Recently Y. Antolin and A. Jaikin-Zapirain have been able to extend this new approach to non-abelian surface groups. Theorem 13.9 ([5]) Let G be a non-abelian surface group. Then for any finitely generated subgroups U and W of G 

d(U ∩ xW x−1 ) ≤ d(U )d(W ).

x∈U \F/W

The strengthened Hanna Neumann conjecture can be also formulated for pro-p groups. The only difference is that we now consider closed subgroups U and W and d(U ) means the number of profinite generators of U . The Howson property for free pro-p groups was proved by A. Lubotzky [74] and the strengthened Hanna Neumann conjecture by A. Jaikin-Zapirain [55]. Again the proof of [55] uses in an essential way the pro-p analogue of the strong Atiyah conjecture for free pro-pgroups (Corollary 11.6). Demushkin pro-p groups are Poincare duality pro-p groups of cohomological dimension 2 and can be seen as pro-p analogues of discrete surface groups. Applying the strategy developed in [55], A. Jaikin-Zapirain and M. Shusterman have proved in [58] the strengthened Hanna Neumann conjecture for non-solvable Demuskin pro-p groups. Theorem 13.10 ([58]) Let G be a non-solvable Demushkin pro-p group. Then for any closed finitely generated subgroups U and W of G 

d(U ∩ xW x−1 ) ≤ d(U )d(W ).

x∈U \F/W

An important step of the proof of the previous theorem is to show that Conjecture 11.1(3) holds for Demushkin pro-p groups over Fp .

Jaikin-Zapirain: L2 -Betti numbers 13.6

399

J.-P. Serre’s problem on torsion-free one-relator pro-p groups

R. Lyndon [82] proved that if a discrete group G is defined by a single relation r = 1, and r is not a power of an element in the free discrete group, then G is of cohomological dimension 2. J.-P. Serre asked whether the analogous statement holds for pro-p groups. D. Gildenhuys [41] found an easy counterexample and reformulated the Serre questions as the following conjecture. Conjecture 13.11 Let G be a finitely generated one-relator torsion-free pro-p group. Then G is of cohomological dimension 2. Proposition 13.12 Let G be a finitely generated one-relator torsion-free pro-p group. Then Conjecture 11.1(3) over Fp implies Conjecture 13.11. Proof Recall that Conjecture 11.1 (1) and (2) hold over Fp by Proposition 11.2. Assume, in addition, that Conjecture 11.1 (3) holds over Fp for G. Let IG be the augmentation ideal of Fp [[G]] and RG the relation module of G. Since G is one-relator group, RG is generated by 1 element. We want to show that RG ∼ = Fp [[G]]. This would imply that G is of cohomological dimension 2. Consider the exact sequence 0 → RG → Fp [[G]]d → IG → 0 (d = d(G)). Since RG = {0}, IG is a proper quotient of Fp [[G]]d . Hence, by Proposition 11.3, dimFp [[G]] IG < d. Since dimFp [[G]] is an integer, dimFp [[G]] IG ≤ d − 1. Thus dimFp [[G]] RG ≥ dimFp [[G]] Fp [[G]]d − dimFp [[G]] IG ≥ d − (d − 1) = 1. Since RG is a quotient of Fp [[G]], applying again Proposition 11.3, we conclude that RG ∼  = Fp [[G]]. 13.7

Properties of the operators in RK[G]

Let K be a subfield of C and G a group. Recall that RK[G] has been defined as the completion of RK[G] with respect to the rkG -metric. If G is countable, we will identify RK[G] with the closure of RK[G] in U (G) with respect to the rkG -metric. In this subsection we discuss the properties of RK[G] and of its elements. In Subsection 10.5 we have already mentioned the strong eigenvalue conjecture (Conjecture 10.12) which can be stated not only for the operators in RK[G] but also for the operators in RK[G] (in fact, this two variations are equivalent). This conjecture was proved in [56] for sofic groups (see also [26] and [110]1 for previous results on this problem). Recall that by Proposition 5.6, if G is an ICC countable group, then Z(RK[G] ) is a subfield of C. Another consequence of Theorem 10.1 is the following result. 1 We warn the reader that the proof of [110, Theorem 4.2] contains a gap, and so, the theorem holds only when A = A∗ . This gap affects also [110, Theorem 4.3].

400

Jaikin-Zapirain: L2 -Betti numbers

Corollary 13.13 ([56]) Let K be a subfield of C closed under complex conjugation and let G be a countable sofic group. Then RK[G] ∩ C = K. In particular, if G is an ICC group, then Z(RK[G] ) = K. For a division ∗-ring D, denote by MD the completion of the direct limit lim Mat2n (D) with respect to the metric induced by its unique Sylvester matrix −→ rank function. If G is ICC, all the known examples of RK[G] are either isomorphic to Matn (D) or to MD for some division ∗-ring D. Moreover, in [33] G. Elek has shown that if H is countable and amenable, then RC[C2 H] is isomorphic to MC . It seems that his proof can be adapted to show that RK[C2 H] is isomorphic to MK for any subfield K of C. Interesting related results have been proved in [6] by P. Ara and J. Claramunt. All this together suggests the following question. Question 13.14 Let G be an ICC group. Is it true that RK[G] is either isomorphic to Matn (D) or to MD for some division ∗-ring D? The next application of Theorem 10.1 shows that the von Neumann rank of a matrix A ∈ Matn×m (K[G]) does not depend on the embedding of K into C if G is sofic. Corollary 13.15 ([56]) Let G be a sofic group. Let K be a field and let φ1 , φ2 : K → C be two embeddings of K into C. Then for every matrix A ∈ Matn×m (K[G]) rkG (φ1 (A)) = rkG (φ2 (A)). Clearly we expect that Corollaries 13.13 and 13.15 are still valid without the assumption that G is sofic. References [1] M. Ab´ert, A. Jaikin-Zapirain and N. Nikolov, The rank gradient from a combinatorial viewpoint, Groups Geom. Dyn. 5 (2011), 213–230. [2] M. Ab´ert and N. Nikolov, Rank gradient, cost of groups and the rank versus Heegaard genus problem, J. Eur. Math. Soc. 14 (2012), 1657–1677. [3] M. Ab´ert, A. Thom and B. Virag, Benjamini-Schramm convergence and pointwise convergence of the spectral measure, preprint, www.renyi.hu/∼abert/luckapprox.pdf. [4] I. Agol, The virtual Haken conjecture. With an appendix by Agol, D. Groves, and J. Manning, Doc. Math. 18 (2013), 1045–1087. [5] Y. Antolin and A. Jaikin-Zapirain, The Hanna Neumann conjecture for hyperbolic limit groups, in preparation. [6] P. Ara and J. Claramunt, Uniqueness of the von Neumann continuous factor, arXiv preprint, arXiv:1705.04501 (2017). [7] P. Ara and K. R. Goodearl, The realization problem for some wild monoids and the Atiyah problem, Trans. Amer. Math. Soc. 369 (2017), 5665–5710. [8] K. Ardakov and K. Brown, Primeness, semiprimeness and localisation in Iwasawa algebras, Trans. Amer. Math. Soc. 359 (2007), 1499–1515.

Jaikin-Zapirain: L2 -Betti numbers

401

[9] M. F. Atiyah, Elliptic operators, discrete groups and von Neumann algebras, Colloque “Analyse et Topologie” en l’Honneur de Henri Cartan (Orsay, 1974), Ast´erisque 32– 33 (Soc. Math. France, Paris 1976), 43–72. [10] T. Austin, Rational group ring elements with kernels having irrational dimension, Proc. Lond. Math. Soc. (3) 107 (2013), 1424–1448. [11] L. Bartholdi, Amenability of groups is characterized by Myhill’s Theorem, with an appendix by D. Kielak, J. Eur. Math. Soc., to appear. [12] S. K. Berberian, Baer ∗-rings, Die Grundlehren der Mathematischen Wissenschaften, Band 195 (Springer-Verlag, New York-Berlin, 1972). [13] S. K. Berberian, The maximal ring of quotients of a finite von Neumann algebra, Rocky Mountain J. Math. 12 (1982), 149–164. [14] N. Bergeron, P. Linnell, W. L¨ uck and R. Sauer, On the growth of Betti numbers in p-adic analytic towers, Groups Geom. Dyn. 8 (2014), 311–329. [15] I. Blomer, P. Linnell and T. Schick, Galois cohomology of completed link groups, Proc. Amer. Math. Soc. 136 (2008), 3449–3459. [16] F. Calegari and M. Emerton, Mod-p cohomology growth in p-adic analytic towers of 3-manifolds, Groups Geom. Dyn. 5 (2011), 355–366. [17] F. Calegari and M. Emerton, Completed cohomology – a survey, Non-abelian fundamental groups and Iwasawa theory (London Math. Soc. Lecture Note Ser., 393, Cambridge Univ. Press, Cambridge, 2012), 239–257. [18] J. Cheeger and M. Gromov, L2 -cohomology and group cohomology, Topology 25 (1986), 189–215. [19] P. M. Cohn, Skew fields. Theory of general division rings, Encyclopedia of Mathematics and its Applications 57 (Cambridge University Press, Cambridge, 1995). [20] P. M. Cohn, Free ideal rings and localization in general rings, New Mathematical Monographs 3 (Cambridge University Press, Cambridge, 2006). [21] W. Dicks, Simplified Mineyev, preprint, http://mat.uab.es/∼dicks. [22] W. Dicks and T. Schick, The spectral measure of certain elements of the complex group ring of a wreath product, Geom. Dedicata 93 (2002), 121–137. [23] J. Dixon, M. du Sautoy, A. Mann and D. Segal, Analytic pro-p groups, Second edition, Cambridge Studies in Advanced Mathematics 61 (Cambridge University Press, Cambridge 1999). [24] J. Dodziuk, de Rham-Hodge theory for L2 -cohomology of infinite coverings, Topology 16 (1977), 157–165. [25] J. Dodziuk and V. Mathai, Approximating L2 -invariants of amenable covering spaces: A combinatorial approach, J. Funct. Anal. 154 (1998), 359–378. [26] J. Dodziuk, P. Linnell, V. Mathai, T. Schick and S. Yates, Approximating L2 invariants and the Atiyah conjecture. Dedicated to the memory of J¨ urgen K. Moser, Comm. Pure Appl. Math. 56 (2003), 839–873. [27] R. M. Dudley, Real analysis and probability, Revised reprint of the 1989 original, Cambridge Studies in Advanced Mathematics 74 (Cambridge University Press, Cambridge 2002). [28] K. Dykema, T. Heister and K. Juschenko, Finitely presented groups related to Kaplansky’s direct finiteness conjecture, Exp. Math. 24 (2015), 326–338. [29] B. Eckmann, Introduction to l2 -methods in topology: reduced l2 -homology, harmonic chains, l2 -Betti numbers. Notes prepared by Guido Mislin, Israel J. Math. 117 (2000), 183–219. [30] G. Elek, The rank of finitely generated modules over group algebras, Proc. Amer. Math. Soc. 131 (2003), 3477–3485. [31] G. Elek, The strong approximation conjecture holds for amenable groups, J. Funct. Anal. 239 (2006), 345–355.

402

Jaikin-Zapirain: L2 -Betti numbers

[32] G. Elek, Connes embeddings and von Neumann regular closures of amenable group algebras, Trans. Amer. Math. Soc. 365 (2013), 3019–3039. [33] G. Elek, Lamplighter groups and von Neumann’s continuous regular ring, Proc. Amer. Math. Soc. 144 (2016), 2871–2883. [34] G. Elek and E. Szab´ o, Sofic groups and direct finiteness, J. Algebra 280 (2004), 426–434. [35] G. Elek and E. Szab´ o, Hyperlinearity, essentially free actions and L2 -invariants. The sofic property, Math. Ann. 332 (2005), 421–441. [36] G. Elek and E. Szab´o, On sofic groups, J. Group Theory 9 (2006), 161–171. [37] G. Elek and E. Szab´o, Sofic representations of amenable groups, Proc. Amer. Math. Soc. 139 (2011), 4285–4291. [38] M. Ershov and W. L¨ uck, The first L2 -Betti number and approximation in arbitrary characteristic, Doc. Math. 19 (2014), 313–332. [39] D. Farkas and P. Linnell, Congruence subgroups and the Atiyah conjecture, Groups, rings and algebras, Contemp. Math. 420 (Amer. Math. Soc., Providence, RI 2006), 89–102. [40] J. Friedman, Sheaves on graphs, their homological invariants, and a proof of the Hanna Neumann conjecture: with an appendix by Warren Dicks, Mem. Amer. Math. Soc. 233 (2014). [41] D. Gildenhuys, On pro-p-groups with a single defining relator, Invent. Math. 5 (1968), 357–366. [42] L

. Grabowski, On Turing dynamical systems and the Atiyah problem, Invent. Math. 198 (2014), 27–69. [43] L

. Grabowski, Group ring elements with large spectral density, Math. Ann. 363 (2015), 637–656. [44] L

. Grabowski, Irrational l2 -invariants arising from the lamplighter group, Groups Geom. Dyn. 10 (2016), 795–817. [45] L

. Grabowski and T. Schick, On computing homology gradients over finite fields, Math. Proc. Cambridge Philos. Soc. 162 (2017), 507–532. [46] K. R. Goodearl, von Neumann regular rings, Second edition, (Robert E. Krieger Publishing Co., Inc., Malabar, FL 1991). [47] K. R. Goodearl and R. B. Warfield, An introduction to noncommutative Noetherian rings, Second edition, London Math. Soc. Student Texts 61 (Cambridge University Press, Cambridge 2004). [48] R. Grigorchuk, P. Linnell, T. Schick and A. Zuk, On a question of Atiyah, C. R. Acad. Sci. Ser I Math. 331 (2000), 663–668. [49] F. Grunewald, A. Jaikin-Zapirain, A. Pinto and P. Zalesskii, Normal subgroups of profinite groups of non-negative deficiency, J. Pure Appl. Algebra 218 (2014), 804– 828. [50] R. Grigorchuk and A. Zuk, The lamplighter group as a group generated by a 2-state automation, and its spectrum, Geom. Dedicata 87 (2001), 209–244. [51] M. Gromov, K¨ ahler hyperbolicity and L2 -Hodge theory, J. Differential Geom. 33 (1991), 263–292. [52] M. Harris, p-adic representations arising from descent on abelian varieties, Compositio Math. 39 (1979), 177–245, Erratum, Compositio Math. 121 (2000), 105–108. [53] B. Hayes and A. Sale, Metric approximations of wreath products, to appear in Ann. Inst. Fourier. [54] A. G. Howson, On the intersection of finitely generated free groups, J. London Math. Soc. 29 (1954), 428–434. [55] A. Jaikin-Zapirain, Approximation by subgroups of finite index and the Hanna Neumann conjecture, Duke Math. J. 166 (2017), 1955–1987.

Jaikin-Zapirain: L2 -Betti numbers

403

[56] A. Jaikin-Zapirain, The base change in the Atiyah and L¨ uck approximation conjectures, preprint, http://verso.mat.uam.es/∼andrei.jaikin/preprints/sac.pdf. ´ [57] A. Jaikin-Zapirain and D. L´opez-Alvarez, On the space of Sylvester matrix rank functions, in preparation. [58] A. Jaikin-Zapirain and M. Shusterman, The Hanna Neumann conjecture for Demushkin groups, in preparation. [59] H. Kammeyer, Introduction to L2 -invariants, course notes available online at https://topology.math.kit.edu/21 679.php. [60] I. Kaplansky, Fields and rings, Reprint of the second (1972) edition, Chicago Lectures in Mathematics (University of Chicago Press, Chicago, IL, 1995). [61] S. Kionke, Characters, L2 -Betti numbers and an equivariant approximation theorem, Arxiv:1702.02599. [62] S. Kionke, The growth of Betti numbers and approximation theorems, Arxiv:1709.00769. [63] A. Knebusch, P. Linnell and T. Schick, On the center-valued Atiyah conjecture for L2 -Betti numbers, Doc. Math. 22 (2017), 659–677. [64] E. Kowalski, Spectral theory in Hilbert spaces, online notes www.math.ethz.ch/∼kowalski/spectral-theory.pdf. [65] P. Kropholler, P. Linnell and J. Moody, Applications of a new K-theoretic theorem to soluble group rings, Proc. Amer. Math. Soc. 104 (1988), 675–684. ´ [66] M. Lazard, Groupes analytiques p-adiques, Inst. Hautes Etudes Sci. Publ. Math. 26 (1965). [67] F. Lehner and S. Wagner, Free lamplighter groups and a question of Atiyah, Amer. J. Math. 135 (2013), 835–849. [68] P. Linnell, Division rings and group von Neumann algebras, Forum Math. 5 (1993), 561–576. [69] P. Linnell, W. L¨ uck and R. Sauer, The limit of Fp -Betti numbers of a tower of finite covers with amenable fundamental groups, Proc. Amer. Math. Soc. 139 (2011), 421– 434. [70] P. Linnell, B. Okun and T. Schick, The strong Atiyah conjecture for right-angled Artin and Coxeter groups, Geom. Dedicata 158 (2012), 261–266. [71] P. Linnell and T. Schick, Finite group extensions and the Atiyah conjecture, J. Amer. Math. Soc. 20 (2007), 1003–1051. [72] P. Linnell and T. Schick, The Atiyah conjecture and Artinian rings, Pure Appl. Math. Q. 8 (2012), 313–327. [73] J. Lott, Heat kernels on covering spaces and topological invariants, J. Differential Geom. 35 (1992), 471–510. [74] A. Lubotzky, Combinatorial group theory for pro-p groups, J. Pure Appl. Algebra 25 (1982), 311–325. [75] A. Lubotzky, A group theoretic characterization of linear groups, J. Algebra 113 (1988), 207–214. [76] W. L¨ uck, Approximating L2 -invariants by their finite-dimensional analogues, Geom. Funct. Anal. 4 (1994), 455–481. [77] W. L¨ uck, The relation between the Baum-Connes conjecture and the trace conjecture, Invent. Math. 149 (2002), 123–152. [78] W. L¨ uck, L2 -invariants: theory and applications to geometry and K-theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics 44 (Springer-Verlag, Berlin, 2002). [79] W. L¨ uck, L2 -invariants from the algebraic point of view, Geometric and cohomological methods in group theory, London Math. Soc. Lecture Note Ser. 358 (Cambridge Univ. Press, Cambridge, 2009), 63–161.

404

Jaikin-Zapirain: L2 -Betti numbers

[80] W. L¨ uck, Estimates for spectral density functions of matrices over C[Zd ], Ann. Math. Blaise Pascal 22 (2015), 73–88. [81] W. L¨ uck, Approximating L2 -invariants by their classical counterparts, EMS Surv. Math. Sci. 3 (2016), 269–344. [82] R. Lyndon, Cohomology theory of groups with a single defining relation, Ann. of Math. (2) 52 (1950), 650–665. [83] P. Malcolmson, Determining homomorphisms to skew fields, J. Algebra 64 (1980), 399–413. [84] J. C. McConnell and J. C.Robson, Noncommutative Noetherian rings. With the cooperation of L. W. Small. Revised edition, Graduate Studies in Mathematics 30 (American Mathematical Society, Providence, RI 2001). [85] I. Mineyev, Submultiplicativity and the Hanna Neumann conjecture, Ann. of Math. (2) 175 (2012), 393–414. [86] J. Moody, Brauer induction for G0 of certain infinite groups, J. Algebra 122 (1989), 1–14. [87] A. Neumann, Completed group algebras without zero divisors, Arch. Math. 51 (1988), 496–499. [88] H. Neumann, On the intersection of finitely generated free groups, Publ. Math. Debrecen 4 (1956), 186–189; Addendum. Publ. Math. Debrecen 5 (1957), 128. [89] W. D. Neumann, On intersections of finitely generated subgroups of free groups, Groups–Canberra 1989 (Lecture Notes in Math., 1456, Springer, Berlin, 1990), 161– 170. [90] D. Northcott and M. Reufel, A generalization of the concept of length, Quart. J. Math. Oxford Ser. (2) 16 (1965), 297–321. [91] D. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Anal. Math. 48 (1987), 1–141. [92] D. Osin, L2 -Betti numbers and non-unitarizable groups without free subgroups, Int. Math. Res. Not. IMRN (2009), 4220–4231. [93] D. Osin, On acylindrical hyperbolicity of groups with positive first l2-Betti number, Bull. Lond. Math. Soc. 47 (2015), 725–730. [94] P. Pansu, Introduction to L2 Betti numbers, Riemannian geometry (Waterloo, ON, 1993), Fields Inst. Monogr., 4 (Amer. Math. Soc., Providence, RI, 1996), 53–86. [95] D. Pape, A short proof of the approximation conjecture for amenable groups, J. Funct. Anal. 255 (2008), 1102–1106. [96] H. D. Petersen, R. Sauer and A. Thom, L2 -Betti numbers of totally disconnected groups and their approximation by Betti numbers of lattices, Arxiv :1612.04559. [97] M. Pichot, T. Schick, and A. Zuk, Closed manifolds with transcendental L2 -Betti numbers, J. London Math. Soc. 92 (2015), 371–392. [98] H. Reich, Group von Neumann Algebras and Related Algebras, Ph.D. Thesis (G¨ottingen 1998), www.mi.fu-berlin.de/math/groups/top/members/publ/diss.pdf. [99] L. Rowen and D. Saltman, Tensor products of division algebras and fields, J. Algebra 394 (2013), 296–309. [100] L. Salce and S. Virili, The addition theorem for algebraic entropies induced by nondiscrete length functions, Forum Math. 28 (2016), 1143–1157. [101] R. Sauer, Power series over the group ring of a free group and applications to NovikovShubin invariants, High-dimensional manifold topology (World Sci. Publ., River Edge, NJ 2003), 449–468. [102] R. Sauer and A. Thom, A spectral sequence to compute L2 -Betti numbers of groups and groupoids, J. Lond. Math. Soc. (2) 81 (2010), 747–773. [103] T. Schick, Integrality of L2 -Betti numbers, Math. Ann. 317 (2000), 727–750, Erratum, Math. Ann. 322 (2002), 421–422.

Jaikin-Zapirain: L2 -Betti numbers

405

[104] T. Schick, L2 -determinant class and approximation of L2 -Betti numbers, Trans. Amer. Math. Soc. 353 (2001), 3247–3265. [105] A. Schofield, Representation of rings over skew fields, London Math. Soc. Lecture Note Ser. 92 (Cambridge University Press, Cambridge, 1985). [106] K. Schreve, The strong Atiyah conjecture for virtually cocompact special groups, Math. Ann. 359 (2014), 629–636. [107] B. Stenstr¨om, Rings of quotients, Die Grundlehren der Mathematischen Wissenschaften, Band 217. An introduction to methods of ring theory (Springer-Verlag, New York-Heidelberg, 1975). [108] P. Symonds and T. Weigel, Cohomology of p-adic analytic groups, New horizons in pro-p groups (Progr. Math., 184, Birkh¨ auser Boston, Boston, MA, 2000), 349–410. [109] N. G. Szoke, Sofic groups, M.Sc. Thesis (E¨ otv¨ os Lor´and University 2014). www.cs.elte.hu/blobs/diplomamunkak/msc mat/2014/szoke nora gabriella.pdf [110] A. Thom, Sofic groups and Diophantine approximation, Comm. Pure Appl. Math. 61 (2008), 1155–1171. [111] P. V´amos, Additive functions and duality over Noetherian rings, Quart. J. Math. Oxford Ser. (2) 19 (1968), 43–55. [112] S. Virili, Crawley-Boewey’s extension of invariants in the non-discrete case, preprint 2017. [113] S. Virili, Algebraic entropy of amenable group actions, preprint 2017. [114] D. Wise, Research announcement: the structure of groups with a quasiconvex hierarchy, Electron. Res. Announc. Math. Sci. 16 (2009), 44–55.

ON THE PRONORMALITY OF SUBGROUPS OF ODD INDEX IN FINITE SIMPLE GROUPS ANATOLY S. KONDRAT’EV∗ , NATALIA MASLOVA , and DANILA REVIN† ∗

Krasovskii Institute of Mathematics and Mechanics UB RAS and Ural Federal University, Yekaterinburg, 620990, Russia Email: [email protected]



Krasovskii Institute of Mathematics and Mechanics UB RAS and Ural Federal University, Yekaterinburg, 620990, Russia Email: [email protected]

† Sobolev Institute of Mathematics SB RAS and Novosibirsk State University, Novosibirsk, 630090, Russia Email: [email protected]

Abstract A subgroup H of a group G is said to be pronormal in G if H and H g are conjugate in H, H g for every g ∈ G. Some problems in finite group theory, combinatorics, and permutation group theory were solved in terms of pronormality. In 2012, E. Vdovin and the third author conjectured that the subgroups of odd index are pronormal in finite simple groups. In this paper we disprove their conjecture and discuss recent progress in the classification of finite simple groups in which the subgroups of odd index are pronormal.

1

Introduction

Throughout the paper we consider only finite groups, and thereby the term “group” means “finite group”. According to P. Hall, a subgroup H of a group G is said to be pronormal in G if H and H g are conjugate in H, H g for every g ∈ G. Some well-known examples of pronormal subgroups are the following: normal subgroups; maximal subgroups; Sylow subgroups; Sylow subgroups of proper normal subgroups; Hall subgroups of solvable groups. In 2012, E. Vdovin and the third author [23] proved that the Hall subgroups are pronormal in all simple groups. The following assertion gives a connection between pronormality of subgroups and properties of permutation representations of finite groups. Theorem 1.1 (P. Hall, 1960s) Let G be a group and H ≤ G. H is pronormal in G if and only if in any transitive permutation representation of G, the subgroup NG (H) acts transitively on the set f ix(H) of fixed points of H. Pronormality is the universal property with respect to Frattini Argument. Indeed, it is not hard to prove the following proposition (see [5, Lemma 4]).

Kondrat’ev, Maslova, Revin: On the pronormality of subgroups of odd index 407 Proposition 1.2 Let G be a group, A  G, and H ≤ A. Then the following statements are equivalent: (1) H is pronormal in G; (2) H is pronormal in A and G = ANG (H); (3) H is pronormal in A and H A = H G . Some problems in finite group theory, combinatorics, and permutation group theory were solved in terms of pronormality. For example, according to L. Babai [1], a group G is called a CI-group if between every two isomorphic relational structures on G (as underlying set) which are invariant under the group GR = {gR | g ∈ G} of right multiplications gR : x → xg (where g, x ∈ G), there exists an isomorphism which is at the same time an automorphism of G. Babai [1] proved that a group G is a CI-group if and only if GR is pronormal in Sym(G). In particular, if G is a CI-group, then G is abelian. With using mentioned Babai’s result, P. Palfy [21] obtained a classification of CI-groups. Thus, the following problem naturally arises. General Problem

Given a finite group G and H ≤ G, is H pronormal in G?

C. Praeger [22] investigated pronormal subgroups of permutation groups. She proved the following theorem. Theorem 1.3 Let G be a transitive permutation group on a set Ω of n points, and let K be a non-trivial pronormal subgroup of G. Suppose that K fixes exactly f points of Ω. Then f ≤ 12 (n − 1), and if f = 12 (n − 1), then K is transitive on its support in Ω, and either G ≥ Alt(n), or G = GLd (2) acting on the n = 2d − 1 non-zero vectors, and K is the pointwise stabilizer of a hyperplane. Thus, if in some permutation representation of G, |f ix(H)| is too big, then H is not pronormal in G. Therefore, it is interesting to consider pronormality of subgroups of a group G containing a subgroup S which is pronormal in G. In particular, it is interesting to consider pronormality of overgroups of Sylow subgroups. Note that the subgroups of odd index in a finite group G are exactly overgroups of Sylow 2-subgroups of G. First of all, we will concentrate on the question of pronormality of subgroups of odd index in non-abelian simple groups. In 2012, E. Vdovin and the third author [23] formulated the following conjecture. Conjecture 1

The subgroups of odd index are pronormal in all simple groups.

In this paper, we disprove Conjecture 1 and discuss recent progress in the classification of non-abelian simple groups in which the subgroups of odd index are pronormal. In Section 11, we discuss the question of pronormality of subgroups of odd index in a non-simple group. Note that an answer to this question for a group G is very weakly connected with answers to this question for subgroups of G, even for

408 Kondrat’ev, Maslova, Revin: On the pronormality of subgroups of odd index normal subgroups of G. For example, it could be proved that there exist infinitely many non-abelian simple groups in which the subgroups of odd index are pronormal while a direct product of any two of them contains a non-pronormal subgroup of odd index (see Example 9.5).

2

Terminology and Notation

Our terminology and notation are mostly standard and can be found in [3, 9]. For a group G and a subset π of the set of all primes, Oπ (G) and Z(G) denote the π-radical (the largest normal π-subgroup) and the center of G, respectively. As usual, π  stands for the set of those primes that do not lie in π. If n is a positive integer, then nπ is the largest divisor of n whose all prime divisors lie in π. Also, for a group G, it is common to write O(G) instead of O2 (G). The set of Sylow p-subgroups of G is denoted by Sylp (G). The socle of G is denoted by Soc(G). Recall that G is almost simple if Soc(G) is a non-abelian simple group. We use the following notation for non-abelian simple groups: Alt(n) for alternating groups; P SLεn (q), where ε = + for P SLn (q) and ε = − for P SUn (q); E6ε (q), where ε = + for E6 (q) and ε = − for 2 E6 (q).  ∞ i i Let m and n be non-negative integers with m = ∞ i=0 ai · 2 and n = i=0 bi · 2 , where ai , bi ∈ {0, 1}. We write m - n if ai ≤ bi for every i and m ≺ n if, in addition, m = n.

3

Verification of Conjecture 1 for many families of non-abelian simple groups

An important role in the verification of Conjecture 1 is played by the following easy assertion (see [23, Lemma 5]), which is a consequence of Theorem 1.1. Lemma 3.1 Suppose that G is a group and H ≤ G. Assume also that H contains a Sylow subgroup S of G. Then the following statements are equivalent: (1) H is pronormal in G; (2) The subgroups H and H g are conjugate in H, H g for every g ∈ NG (S). Note that Sylow 2-subgroups in non-abelian simple groups are usually selfnormalized, and all the exceptions were described by the first author in [10]. He proved the following theorem (see [10, Corollary of Theorems 1-3]). Theorem 3.2 Let G be a non-abelian simple group and let S ∈ Syl2 (G). Then NG (S) = S excluding the following cases: (1) G ∼ = J2 , J3 , Suz or F5 and |NG (S) : S| = 3; (2) G ∼ = 2 G2 (32n+1 ) or J1 and NG (S) ∼ = (C2 )3  (C7  C3 ) < Hol((C2 )3 );

(3) G is a group of Lie type over a field of characteristic 2 and NG (S) is a Borel subgroup of G; (4) G ∼ = P SL2 (q), where 3 < q ≡ ±3 (mod 8) and NG (S) ∼ = Alt(4);

Kondrat’ev, Maslova, Revin: On the pronormality of subgroups of odd index 409 (5) G ∼ = P Sp2n (q), where n ≥ 2, q ≡ ±3 (mod 8), n = 2s1 + · · · + 2st for s1 > · · · > st ≥ 0 and NG (S)/S is the elementary abelian group of order 3t ; η (6) G ∼ = E (q) where η ∈ {+, −}, q is odd, and |NG (S) : S| = (q − η1)2 /(q − 6

η1, 3) = 1; η (7) G ∼ = P SLn (q), where n ≥ 3, η ∈ {+, −}, q is odd, n = 2s1 + · · · + 2st for s1 > · · · > st ≥ 0, and NG (S) ∼ = S × C1 × · · · × Ct−1 > S, where C1 , . . . , Ct−2 and Ct−1 are cyclic subgroups of orders (q − η1)2 , . . . , (q − η1)2 and (q − η1)2 /(q − η1, n)2 , respectively.

Thus, using Lemma 3.1 for the verification of Conjecture 1, it is sufficient to consider simple groups from items (1)–(7) of Theorem 3.2. In [11] we considered simple groups from items (1)–(4) and, with taking into account of simple groups with self-normalized Sylow 2-subgroups, we have proved the following theorem. Theorem 3.3 The subgroups of odd index are pronormal in the following simple groups: Alt(n), where n ≥ 5; sporadic groups; groups of Lie type over fields of characteristic 2; P SLε2n (q), where ε ∈ {+, −}; P Sp2n (q), where q ≡ ±3 (mod 8); orthogonal groups; exceptional groups of Lie type not isomorphic to E6ε (q), where ε ∈ {+, −}.

4

Maximal subgroups of odd index in simple groups

In this paper, the complete classification of maximal subgroups of odd index in simple classical groups is a crucial tool in view of the following evident lemma. Lemma 4.1 Suppose that H and M are subgroups of a group G and H ≤ M . Then (1) if H is pronormal in G, then H is pronormal in M ; (2) if S ≤ H for some Sylow subgroup S of G, NG (S) ≤ M , and H is pronormal in M , then H is pronormal in G. M. Liebeck and J. Saxl [16] and, independently, W. Kantor [7] proposed a classification of primitive permutation groups of odd degree. It is considered to be one of remarkable results in the theory of finite permutation groups. In particular, both papers [16] and [7] contain lists of subgroups of simple groups that can turn out to be maximal subgroups of odd index. However, in the cases of alternating groups and of classical groups over fields of odd characteristics, neither in [16] nor in [7] it was described which of the specified subgroups are precisely maximal subgroups of odd index. Thus, the problem of the complete classification of maximal subgroups of odd index in simple groups was remained open. The classification was finished by the second author in [18, 20]. In [18], the author referred to results obtained by P. Kleidman [8] and by P. Kleidman and M. Liebeck [9]. However, there are a number of inaccuracies in Kleidman’s PhD thesis [8]. These inaccuracies have been corrected in [2]. Due to uncovered circumstances, in [19] the second author revised the main result of [18] taking into account the

410 Kondrat’ev, Maslova, Revin: On the pronormality of subgroups of odd index results of [2]. In particular, changes in statements of items (6), (10), and (21) of [18, Theorem 1] were made.

5

Some tools for disproving Conjecture 1

In [12], we disproved Conjecture 1. The aim of this section is to demonstrate some tools for this. A consequence of well-known Schur–Zassenhaus theorem (see, for example, [6, Theorems 3.8 and 3.12]) is the following proposition [6, Ch. 4, Lemma 4.28]. Proposition 5.1 If V is a normal subgroup of a group G and H is a subgroup of G such that (|H|, |V |) = 1, then, for any H-invariant subgroup U of V , the equality U = CU (H)[H, U ] holds. We proved that the following more general statement [12, Proposition 2] holds. Proposition 5.2 If V is a normal subgroup of a group G and H is a pronormal subgroup of G, then, for any H-invariant subgroup U of V , the equality U = NU (H)[H, U ] holds. It is easy to see that, in the case when the subgroups H and V from Proposition 5.2 have trivial intersection, the equality NU (H) = CU (H) holds for any H-invariant subgroup U of V . Therefore, Proposition 5.1 is a special case of Proposition 5.2. We showed that the statement converse to Proposition 5.2 holds when the group V is abelian and G = HV (i.e., H is a supplement to the subgroup V in G). We have proved the following theorem (see [12, Theorem 1]). Theorem 5.3 Let H and V be subgroups of a group G such that V is an abelian normal subgroup of G and G = HV . Then the following statements are equivalent: (1) the subgroup H is pronormal in G; (2) U = NU (H)[H, U ] for any H-invariant subgroup U of V . With using Theorem 5.3, we have proved the following proposition (see [12, Corollary of Theorem 1]). Proposition 5.4 Let G = A  Sym(n) = HV be the natural permutational wreath product of an abelian group A and the symmetric group H = Sym(n), where V denotes the base of the wreath product. Then the following statements are equivalent: (1) the subgroup H is pronormal in G; (2) (|A|, n) = 1.

Kondrat’ev, Maslova, Revin: On the pronormality of subgroups of odd index 411

6

A series of examples disproving Conjecture 1

The aim of this section is to construct a series of examples disproving Conjecture 1. We prove the following theorem (see [12, Theorem 2]). Theorem 6.1 The simple group P Sp6n (q) for any q ≡ ±3 (mod 8) contains a non-pronormal subgroup of odd index. Sketch of proof Let q ≡ ±3 (mod 8) be a prime power and n be a positive integer. It is well known that a Sylow 2-subgroup S of the group T = Sp2 (q) = SL2 (q) is isomorphic to Q8 , and NT (S) ∼ = SL2 (3) ∼ = Q8 : 3. We have the following chain of embeddings: Q8  Sym(3n) ≤ L = Sp2 (3)  Sym(3n) ≤ Sp2 (q)  Sym(3n) ≤ G = Sp6n (q). It could be proved by direct calculations that |G : L| is odd. It is easy to see that L/O2 (L) ∼ = C3  Sym(3n). In view of Proposition 5.4, the group L/O2 (L) contains a non-pronormal subgroup R of odd index. Let H be the preimage of R in L. Then H is a non-pronormal subgroup of odd index in L in view of the following proposition. Proposition 6.2 (see [17, Lemma 3] and [4, Chapter I, Proposition (6.4)]) Suppose that H is a subgroup and N is a normal subgroup of a group G. Let ¯ : G → G/N be the natural epimorphism. The following statements hold: (1) if H is pronormal in G, then H is pronormal in G; (2) H is pronormal in G if and only if HN is pronormal in G and H is pronormal in NG (HN ); (3) if N ≤ H and H is pronormal in G, then H is pronormal in G. In particular, a subgroup H of odd index is pronormal in G if and only if H/O2 (G) is pronormal in G/O2 (G). In view of Lemma 4.1, H is a non-pronormal subgroup of G, and it is easy to see that |G : H| is odd. Note that O2 (G) ≤ H. Thus, H/O2 (G) is a non-pronormal subgroup of odd index in G/O2 (G) ∼ = P Sp6n (q) in view of Proposition 6.2.

7

Classification problem

The following problem naturally arises. Problem 1 Classify non-abelian simple groups in which the subgroups of odd index are pronormal. To solve Problem 1 it remains to consider the following simple groups: (1) P Sp2n (q), where q ≡ ±3 (mod 8) and 3 does not divide n; (2) E6ε (q), where ε ∈ {+, −} and q is odd;

412 Kondrat’ev, Maslova, Revin: On the pronormality of subgroups of odd index (3) P SLεn (q), where ε ∈ {+, −}, q is odd, and n = 2w . In this paper, we consider in some details a solution of Problem 1 for symplectic groups, briefly discuss a solution of Problem 1 for groups E6ε (q), and formulate a conjecture for groups P SLεn (q).

8

Simple symplectic groups containing non-pronormal subgroups of odd index

In fact, Theorem 6.1 permits us to investigate Problem 1 within a much wider family of symplectic groups. Let G = Sp2n (q), where q is odd, and V be the natural 2n-dimensional vector space over the field Fq with a non-degenerate skew-symmetric bilinear form associated with G. Let us look to the list of maximal subgroups of odd index of G (this list can be found in [19]). Proposition 8.1 Maximal subgroups of odd index in Sp2n (q) = Sp(V ), where n > 1 and q is odd, are the following: (1) Sp2n (q0 ), where q = q0r and r is an odd prime; (2) Sp2m (q) × Sp2(n−m) (q), where m ≺ n; (3) Sp2m (q)  Sym(t), where n = mt and m = 2k ; (4) 21+4 + .Alt(5), where n = 2 and q ≡ ±3 (mod 8) is a prime. Note that if q is odd, then |Z(Sp2n (q))| = 2. Thus, in view of Proposition 6.2, the subgroups of odd index are pronormal in Sp2n (q) if and only if the subgroups of odd index are pronormal in P Sp2n (q), i.e., there is no difference between investigation of Problem 1 for Sp2n (q) or for P Sp2n (q). Let q ≡ ±3 (mod 8) and n is not of the form 2w or 2w (22k + 1). Then the ∞ 2-adic decomposition n = i=0 si · 2i either has two 1s in positions s1 and s2 of different parity, or three 1s in positions s1 , s2 , and s3 of the same parity. Define m = 2s1 + 2s2 or m = 2s1 + 2s2 + 2s3 , respectively. It is easy to see that m ≺ n and 3 divides m. Let M be the stabilizer in G of a non-degenerate 2m-dimensional subspace of V . It is easy to see that M = M1 × M2 , where M1 ∼ = Sp2m (q) and M2 ∼ = Sp2(n−m) (q). Note that the index |G : M | is odd by Proposition 8.1. Thus, if H is a subgroup of odd index in M , then H is a subgroup of odd index in G too. Since 3 divides m, it follows by Theorem 6.1 that M1 /Z(M1 ) = M1 /O2 (M1 ) has a non-pronormal subgroup H1 /O2 (M1 ) of odd index. Then H1 is a non-pronormal subgroup of odd index in M1 in view of Proposition 6.2. Now it is easy to see that therefore, H1 ×M2 is a non-pronormal subgroup of odd index in M1 × M2 . So, H1 × M2 is a nonpronormal subgroup of odd index in G. Thus, we proved the following theorem (see [13, Theorem 1]). Theorem 8.2 Let G = P Sp2n (q), where q ≡ ±3 (mod 8) and n is not of the form 2w or 2w (22k + 1). Then G has a non-pronormal subgroup of odd index.

Kondrat’ev, Maslova, Revin: On the pronormality of subgroups of odd index 413 Now in view of Theorems 3.3 and 8.2, to finish a classification of simple symplectic groups in which the subgroups of odd index are pronormal, it remains to consider groups P Sp2n (q), where q ≡ ±3 (mod 8) and n is of the form 2w or 2w (22k + 1). In the next section we will prove that the subgroups of odd index are pronormal in these groups.

9

Classification of simple symplectic groups in which the subgroups of odd index are pronormal

In this section we prove the following theorem, whose proof was recently finished by the authors (see [13, Theorem 2] for the case when n is of the form 2w and [14] for the case when n is of the form 2w (22k + 1)). Theorem 9.1 Let G = P Sp2n (q). Then each subgroup of odd index is pronormal in G if and only if one of the following statements holds: (1) q ≡ ±3 (mod 8); (2) n is of the form 2w or 2w (22k + 1). Sketch of proof Let G = Sp2n (q), where q ≡ ±3 (mod 8) and n is of the form 2w ≥ 2. Suppose that the claim of the theorem is false, and let q be the smallest prime power congruent to ±3 modulo 8 such that G has a non-pronormal subgroup H of odd index. Let S ≤ H be a Sylow 2-subgroup of G. Note that |NG (S)/S| = 3 in view of Theorem 3.2. Pick g ∈ NG (S). Without loss of generality we can assume that |g| = 3. Put K = H, H g , and we can suppose that K < G. Then there exists a maximal subgroup M of G such that K ≤ M . Using Proposition 8.1, we conclude that one of the following cases arises. 1. M ∼ = Sp2n (q0 ), where q = q0r and r is an odd prime. In this case it could be proved that NG (S) ≤ M , and we use inductive reasonings and Lemma 4.1 to prove that H is pronormal in G. 2. M ∼ = Sp2m (q)  Sym(t), where n = mt and m = 2k , and M is choosen of such type so that m is as small as possible. In this case we can prove that H/O2 (M ) is pronormal in M/O2 (M ) ∼ = P Sp2m (q)Sym(t) with using two following propositions and additional inductive reasonings. Proposition 9.2 (see [13, Lemma 15] and [5, Lemma 9]) Let Q be a subgroup of odd index in a group L = L1 × L2 × . . . × Ln , where Li are groups, and let πi be the projection from L to Li . If there is i such that Li is almost simple, Li /Soc(Li ) is a 2-group, and Qπi = Li , then Li ≤ Q. Proposition 9.3 (see [13, Lemma 17]) Let G = A  Sym(n) = LH be the natural permutation wreath product of a non-abelian simple group A and H = Sym(n), where n = 2w , L = L1 × . . . × Ln , and for each i ∈ {1, . . . , n}, Li ∼ = A and πi : L → Li is the projection from L to Li . If K is a subgroup of odd index in G, K0 = K ∩L, and M1 ≤ L1 such that NL1 (K0π1 ) ≤ M1 , then K ≤ U ∼ = M1  Sym(n) ≤ G.

414 Kondrat’ev, Maslova, Revin: On the pronormality of subgroups of odd index Moreover, it could be proved that in this case NG (S) ≤ M . Now, in view of Proposition 6.2 and Lemma 4.1, we conclude that H is pronormal in G. 3. M ∼ = 21+4 + .Alt(5), where n = 4 and q is a prime. In this case it could be easy proved that NG (S) ≤ M . Now we conclude that H is pronormal in G in view of Proposition 6.2, Lemma 4.1, and Theorem 3.3. In the case when n is of the form 2w (22k + 1), the scheme of the proof is similar, however we can not use Proposition 9.3. The proof of Proposition 9.3 is based on a fact that K contains a regular subgroup of some conjugate of H. And the fact could be false if n is not of the form 2w . To investigate this case the second and the third authors in a joint work with W. Guo [5] obtained the following useful criterion of the pronormality of subgroups of odd index in extensions of groups (see [5, Theorem 1]). Theorem 9.4 Let G be a group, AG, the subgroups of odd index are pronormal in A, and Sylow 2-subgroups of G/A are self-normalized. Let T be a Sylow 2-subgroup of A. Then the following statements are equivalent: (1) The subgroups of odd index are pronormal in G; (2) The subgroups of odd index are pronormal in NG (T )/T . However, there are difficulties in a direct application of Theorem 9.4 to the maximal subgroups of the group Sp2n (q), where n is of the form 2w (22k + 1), in view of the following fact: if the subgroups of odd index are pronormal in both groups G1 and G2 , it does not imply that the subgroups of odd index are pronormal in the group G1 × G2 , even if G1 and G2 are non-abelian simple. It is clear from the following example (see [5, Proposition 1]). Example 9.5 Consider Frobenius groups Hi = Li Ki ∼ = C7 C3 for i ∈ {1, 2} and H = H1 ×H2 . Note that any proper subgroup of Hi is its Sylow subgroup, and hence is pronormal in Hi . Let L = O7 (H) = L1 ×L2 ∼ = C7 ×C7 and D = {(x, x) | x ∈ C7 }. There exists k1 ∈ K1 ×{1} such that Dk1 = {(2x, x) | x ∈ C7 }. Hence D, Dk1 = L is abelian and D is a non-pronormal subgroup (of odd index) in H. Let G1 , G2 ∈ {J1 } ∪ {2 G2 (32m+1 ) | m ≥ 1} and Si ∈ Syl2 (Gi ). In view of Theorem 3.3, the subgroups of odd index are pronormal in Gi for i ∈ {1, 2}. And in view of Theorem 3.2, NGi (Si )/Si for i ∈ {1, 2} is isomorphic to the Frobenius group C7  C3 . Using previous reasonings it is easy to construct a non-pronormal subgroup of odd index in G = G1 × G2 . Recently basing on Theorem 9.4, the second and the third authors, and W. Guo [5] obtained the following pronormality criterion for subgroups of odd index in  groups of the type ti=1 (A  Sym(ni )), where A is an abelian group and all the wreath products are natural permutation (see [5, Theorem 2]).  Theorem 9.6 Let A be an abelian group and G = ti=1 (A  Sym(ni )), where all the wreath products are natural permutation. Then the subgroups of odd index are pronormal in G if and only if for any positive integer m, the inequality m ≺ ni for some i implies that g.c.d.(|A|, m) is a power of 2.

Kondrat’ev, Maslova, Revin: On the pronormality of subgroups of odd index 415 Moreover, in [5, Theorem 3] the following theorem was proved.  Theorem 9.7 Let G = ti=1 Gi , where for any i ∈ {1, . . . , t}, Gi ∼ = Sp2ni (qi ), qi is odd, and ni is a power of 2. Then the subgroups of odd index are pronormal in G. Theorems 9.4, 9.6, and 9.7 became the main tools in the proof of Theorem 9.1 which was finished by the authors in [14].

10

Summary and further research on finite simple groups in which the subgroups of odd index are pronormal

Let G be a non-abelian simple group, S ∈ Syl2 (G), and C = O(CG (S)). In [15, Theorem 7] it was proved that C = 1 if and only if one of the following statements holds: ∼ E η (q), where η ∈ {+, −}, q is odd, and C is a cyclic group of order (1) G = 6

(q − η1)2 /(q − η1, 3) = 1; η (2) G ∼ = P SLn (q), where n ≥ 3, η ∈ {+, −}, q is odd, n = 2s1 + · · · + 2st for s1 > · · · > st ≥ 0 and t > 1, and C ∼ = C1 × · · · × Ct−1 = 1, where C1 , . . . Ct−2 , and Ct−1 are cyclic subgroups of orders (q − η1)2 , . . . , (q − η1)2 , and (q − η1)2 /(q − η1, n)2 , respectively.

Thus, with taking into account of Theorems 3.3 and 9.1, we obtain the following theorem. Theorem 10.1 Let G be a non-abelian simple group, S ∈ Syl2 (G), and C = O(CG (S)). If C = 1, then exactly one of the following statements holds: (1) The subgroups of odd index are pronormal in G; (2) G ∼ = P Sp2n (q), where q ≡ ±3 (mod 8) and n is not of the form 2w or w 2 (22k + 1). Moreover, recently we have obtained a solution of Problem 1 for simple exceptional groups E6ε (q). We have proved the following theorem. Theorem 10.2 Let G = E6ε (q), where q = pk , p is a prime, and ε ∈ {+, −}. Then the subgroups of odd index are pronormal in G if and only if 18 does not divide q − ε1 and if p is odd and ε = +, then k is a power of 2. The scheme of our proof of Theorem 10.2 is similar as for symplectic groups and is based on the classification of maximal subgroups of odd index in simple exceptional groups of Lie type obtained in [16, 7]. Moreover, we use some results on subgroup structure of G = E6ε (q) obtained in [15]. The most difficult case here is when a possibly non-pronormal subgroup H of odd index and its conjugate H g for some g ∈ G are both contained in a parabolic maximal subgroup of G. In this case Theorem 9.4 is an useful tool. Thus, to solve Problem 1 it remains to consider the groups P SLεn (q), where ε ∈ {+, −}, q is odd, and n = 2w . We have the following conjecture.

416 Kondrat’ev, Maslova, Revin: On the pronormality of subgroups of odd index Conjecture 2 Let G = P SLεn (q), where q is odd and ε ∈ {+, −}. The subgroups of odd index are pronormal in G if and only if for any positive integer m, the inequality m ≺ n implies that g.c.d.(m, q 1+ε1 (q − ε1)) is a power of 2.

11

Question of the pronormality of subgroups of odd index in non-simple groups

The Frattini Argument (see Proposition 1.2) and Proposition 6.2 are convenient tools to reduce General Problem to groups of smaller order. Assume that G is not simple and A is a minimal non-trivial normal subgroup of G. Then A is a direct product of pairwise isomorphic simple groups, and one of the following cases arises: (1) A ≤ H and, in view of Proposition 6.2, H is pronormal in G if and only if H/A is pronormal in G/A. Note that |G/A| < |G|. (2) H ≤ A and, in view of Proposition 1.2, H is pronormal in G if and only if H is pronormal in A and G = ANG (H). Note that |A| < |G|. Thus, the question of pronormality of subgroups in direct products of simple groups is of interest. (3) Let H ≤ A and A ≤ H, and let N = NG (HA). In view of Propositions 6.2 and 1.2, H is pronormal in G if and only if HA/A is pronormal in G/A, N = ANN (H), and H is pronormal in HA. Therefore, with using inductive reasonings, we can reduce this case to the subcase when G = HA. Suppose that G = HA, A is a minimal non-trivial normal subgroup of G, A ≤ H, and |G : H| is odd. If |A| is odd, then A is abelian and H is pronormal in G. Indeed, if U is an H-invariant subgroup of A, then either U is trivial or U = A. Therefore, A ∩ H is trivial, and it is easy to see that U = CU (H)[H, U ] = NU (H)[H, U ] for every H-invariant subgroup U of A. Thus, in view of Theorem 5.3, H is pronormal in G. Note that A is not a 2-group, because we suggested that A ≤ H. Suppose that A is a direct product of pairwise isomorphic non-abelian simple groups. If the subgroups of odd index are pronormal in A, then we can use the following criterion of the pronormality of subgroups of odd index in extensions of groups, which is a generalization of Theorem 9.4. Theorem 11.1 Let G be a group, A  G, and the subgroups of odd index are pronormal both in A and in G/A. Let T be a Sylow 2-subgroup of A and let ¯ : G → G/A be the natural epimorphism. (1) Assume that H ≤ G, S ≤ H for some S ∈ Syl2 (G), and T = A ∩ S. Let Y = NA (H ∩ A) and Z = NH∩A (T ). Then H is pronormal in G if and only if NH (T )/Z is pronormal in (NH (T )NY (T ))/Z and NG (H) = NG (H). (2) Let T ∈ Syl2 (A). The subgroups of odd index are pronormal in G if and only if the subgroups of odd index are pronormal in NG (T )/T and for every subgroup H ≤ G of odd index, we have NG (H) = NG (H). Note that proof of Theorem 11.1 follows from proofs of Theorems 1 and 4 in [5].

Kondrat’ev, Maslova, Revin: On the pronormality of subgroups of odd index 417 In view of Theorem 11.1, the following problem is of interest. Problem 2 Describe direct products of non-abelian simple groups in which the subgroups of odd index are pronormal. Note that Problem 2 is not equivalent to Problem 1 (see Example 9.5). However, in some cases the pronormality of subgroups of odd index in a direct product of non-abelian simple groups is equivalent to the pronormality of subgroups of odd index in each component. As an example, recently we have proved the following theorem.  Theorem 11.2 Let G = ti=1 Gi , where Gi ∼ = P Sp2ni (qi ) and qi is odd for each i ∈ {1, ..., t}. Then the subgroups of odd index are pronormal in G if and only if the subgroups of odd index are pronormal in Gi for each i. References [1] Laszlo Babai, Isomorphism problem for a class of point-symmetric structures, Acta Math. Acad. Sci. Hungar. 29 (1977), 329–336. [2] John N. Bray, Derek F. Holt, Colva M. Roney-Dougal, The Maximal Subgroups of the Low-Dimensional Finite Classical Groups (CUP, Cambridge 2013). [3] John H. Conway, Robert T. Curtis, Simon P. Norton, Richard A. Parker and Robert A. Wilson, Atlas of Finite Groups (OUP, Oxford 1985). [4] Klaus Doerk, Trevor O. Hawkes, Finite Soluble Groups (Walter de Gruyter, Berlin and New York 1992). [5] Wenbin Guo, Natalia V. Maslova and Danila O. Revin, On the pronormality of subgroups of odd index in some extensions of finite groups, Siberian Math. J. 59:4 (2018), to appear. [6] Martin I. Isaacs, Finite Group Theory (Amer. Math. Soc., Providence, RI 2008). [7] William. M. Kantor, Primitive permutation groups of odd degree, and an application to finite projective planes, J. Algebra. 106:1 (1987), 15–45. [8] Peter Kleidman, The Subgroup Structure of Some Finite Simple Groups, Ph.D. Thesis (Cambridge Univ., Cambridge 1986). [9] Peter B. Kleidman and Martin Liebeck, The Subgroup Structure of the Finite Classical Groups (CUP, Cambridge 1990). [10] Anatoly S. Kondrat’ev, Normalizers of the Sylow 2-subgroups in finite simple groups, Math. Notes 78:3 (2005), 338–346. [11] Anatoly S. Kondrat’ev, Natalia V. Maslova and Danila O. Revin, On the pronormality of subgroups of odd index in finite simple groups, Siberian Math. J. 56:6 (2015), 1001– 1007. [12] Anatoly S. Kondrat’ev, Natalia V. Maslova and Danila O. Revin, A pronormality criterion for supplements to abelian normal subgroups, Proc. Steklov Inst. Math. 296:Suppl. 1 (2017), 1145–1150. [13] Anatoly S. Kondrat’ev, Natalia V. Maslova and Danila O. Revin, On the pronormality of subgroups of odd index in finite simple symplectic groups, Siberian Math. J. 58:3 (2017), 467–475. [14] Anatoly S. Kondrat’ev, Natalia V. Maslova and Danila O. Revin, On pronormal subgroups in finite simple groups, Dokl. Math. 98:2 (2018), 1–4. [15] Anatoly S. Kondratiev and Viktor D. Mazurov, 2-Signalizers of finite simple groups, Algebra Logic 42:5 (2003), 333–348.

418 Kondrat’ev, Maslova, Revin: On the pronormality of subgroups of odd index [16] Martin W. Liebeck and Jan Saxl, The primitive permutation groups of odd degree, J. London Math. Soc. (2) 31:2 (1985), 250–264. [17] Victor D. Mazurov, 2-Signalizers of finite groups, Algebra Logic 7:3 (1968), 167–168. [18] Natalia V. Maslova, Classification of maximal subgroups of odd index in finite simple classical groups, Proc. Steklov Inst. Math. 267:Suppl. 1 (2009), 164–183. [19] Natalia V. Maslova, Classification of maximal subgroups of odd index in finite simple ` classical groups: Addendum, Sib. Electron. Mat. Izv. 15 (2018), 707–718. [20] Natalia V. Maslova, Classification of maximal subgroups of odd index in finite groups with alternating socle, Proc. Steklov Inst. Math. 285:Suppl. 1 (2014), 136–138. [21] Peter P. Palfy, Isomorphism problem for relational structures with a cyclic automorphism, European J. Combin. 8 (1987), 35–43. [22] Cheryl E. Praeger, On transitive permutation groups with a subgroup satisfying a certain conjugacy condition, J. Austral. Math. Soc. 36:1 (1984), 69–86. [23] Evgeny P. Vdovin and Danila O. Revin, Pronormality of Hall subgroups in finite simple groups, Siberian Math. J. 53:3 (2012), 419–430.

VERTEX STABILIZERS OF GRAPHS WITH PRIMITIVE AUTOMORPHISM GROUPS AND A STRONG VERSION OF THE SIMS CONJECTURE ANATOLY S. KONDRAT’EV∗ and VLADIMIR I. TROFIMOV† Krasovskii Institute of Mathematics and Mechanics of UB RAS and Ural Federal University, Ekaterinburg, 620990, Russia ∗ Email: [email protected] † Email: trofi[email protected]

Abstract The well-known Sims conjecture was proved by P. Cameron, C. Praeger, J. Saxl and G. Seitz in 1983. We survey our results proving a strengthened version of the Sims conjecture and also our results aimed at obtaining an even stronger version of the Sims conjecture.

1

Sims conjecture and its strengthened version

Let G be a primitive permutation group on a finite set X and x ∈ X. Let d be the length of some Gx -orbit on X \{x}. It is easy to see that d = 1 implies Gx = 1 (and G∼ = Z2 (and G ∼ = Zp for a prime p) and d = 2 implies Gx ∼ = D2p for an odd prime p). In [21], C. Sims adapted arguments by W. Tutte concerning vertex stabilizers of cubic graphs in vertex-transitive groups of automorphisms (see [23, 24]) to prove that d = 3 implies |Gx | divides 3 · 24 . In connection with this result C. Sims made the following general conjecture which is now well known as the Sims conjecture. Sims conjecture There exists a function f : N −→ N such that, if G is a primitive permutation group on a finite set X, Gx is the stabilizer in G of a point x from X, and d is the length of some nontrivial Gx -orbit on X \ {x}, then |Gx | ≤ f (d). Some progress toward proving this conjecture had been obtained in [21, 22, 25, 13, 8] and some other papers. In particular, J. Thompson [22] and independently H. Wielandt [25] proved that |Gx /Op (Gx )| is bounded by a function of d for some prime p. But only using the classification of finite simple groups was the validity of the conjecture proved by P. Cameron, C. Praeger, J. Saxl and G. Seitz in [4]. This proof implies that one can take a function of the form exp(Cd3 ), where C is some constant, as the function f (d) in the Sims conjecture. The Sims conjecture can be formulated using graphs as follows. For an undirected connected graph Γ (without loops or multiple edges) with [i] vertex set V (Γ), G ≤ Aut(Γ), x ∈ V (Γ), and i ∈ N ∪ {0}, we denote by Gx the The work is supported by the Russian Science Foundation (grant no. 14-11-00061-P).

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elementwise stabilizer in G of the ball of radius i of the graph Γ centered at x in the natural metric on V (Γ). Let G be a primitive permutation group on a finite set X and x, y ∈ X, x = y. Let ΓG,{x,y} be the graph with vertex set V (ΓG,{x,y} ) = X and edge set E(ΓG,{x,y} ) = {{g(x), g(y)} | g ∈ G}. Then ΓG,{x,y} is an undirected connected graph, G is an automorphism group of ΓG,{x,y} acting primitively on V (ΓG,{x,y} ), and the length of the Gx -orbit containing y is equal either to the valency of ΓG,{x,y} (if there exists an element in G that transposes x and y) or to the half of the valency of ΓG,{x,y} (otherwise). Now it is easy to see that the Sims conjecture can be reformulated in the following form. Sims conjecture (geometrical form) There exists a function ψ : N∪{0} −→ N such that, if Γ is an undirected connected finite graph and G is its automorphism [ψ(d)] group acting primitively on V (Γ), then Gx = 1 for x ∈ V (Γ), where d is the valency of the graph Γ. Using the classification of finite simple groups, we obtained in [14] the following result, which establishes the validity of a strengthened version of the Sims conjecture (in the geometrical form). Theorem 1.1 If Γ is an undirected connected finite graph and G is its automor[6] phism group acting primitively on V (Γ), then Gx = 1 for x ∈ V (Γ). Theorem 1.1 says, in other words, that automorphisms of connected finite graphs with vertex-primitive automorphism groups are determined by images of vertices of any ball of radius 6. (It can be also said that a ball of radius 6 is a base for G.) Actually in [14], we obtained a result (see Theorem 1.2 below) which implies Theorem 1.1 and concerns the subgroup structure of finite groups. To formulate the result, we need the following definitions.  As usual, for a group G and H ≤ G, we put HG = g∈G gHg −1 (the core of the subgroup H in G). For a group G and its subgroups M1 and M2 , let us define inductively on i ∈ N subgroups (M1 , M2 )i and (M2 , M1 )i of M1 ∩M2 , which will be called the ith mutual cores of M1 with respect to M2 and of M2 with respect to M1 , respectively. Put (M1 , M2 )1 = (M1 ∩ M2 )M1 ,

(M2 , M1 )1 = (M1 ∩ M2 )M2 .

For i ∈ N, assuming that (M1 , M2 )i and (M2 , M1 )i are defined, put (M1 , M2 )i+1 = ((M2 , M1 )i )M1 ,

(M2 , M1 )i+1 = ((M1 , M2 )i )M2 .

Mutual cores of subgroups M1 and M2 of a group G have the following obvious properties. For i ∈ N, the equality (M1 , M2 )i = (M2 , M1 )i means that

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421

this subgroup is maximal in M1 ∩ M2 with respect to the property of being normal both in M1 and in M2 , and all the groups (M1 , M2 )i+j and (M2 , M1 )i+j for j ∈ N coincide with it. For k ∈ {1, 2} and i ∈ N, the equality (Mk , M3−k )i+1 = (Mk , M3−k )i is equivalent to the inclusion (Mk , M3−k )i ≤ (M3−k , Mk )i , and the equality (Mk , M3−k )i+2 = (Mk , M3−k )i implies the equality (M1 , M2 )i+1 = (M2 , M1 )i+1 . If G is a primitive permutation group on a finite set X and x, y ∈ X, x = y, then we have the following interpretation of mutual cores (Gx , Gy )i and (Gy , Gx )i for i ∈ N. Let ΓG,(x,y) be the directed graph with V (ΓG,(x,y) ) = X and E(ΓG,(x,y) ) = {(g(x), g(y)) | g ∈ G}, i.e., the directed graph corresponding to the orbital of G containing (x, y). Then (Gx , Gy )i is the pointwise stabilizer in Gx of the set {z ∈ V (ΓG,(x,y) ) | there exist 0 ≤ j ≤ i and z0 , . . . , zj ∈ V (ΓG,(x,y) ) such that z0 = x, zj = z, (zk , zk+1 ) ∈ E(ΓG,(x,y) ) for all even 0 ≤ k < j and (zk+1 , zk ) ∈ E(ΓG,(x,y) ) for all odd 0 < k < j} and (Gy , Gx )i is the pointwise stabilizer in Gy of the set {z ∈ V (ΓG,(x,y) ) | there exist 0 ≤ j ≤ i and z0 , . . . , zj ∈ V (ΓG,(x,y) ) such that z0 = y, zj = z, (zk+1 , zk ) ∈ E(ΓG,(x,y) ) for all even 0 ≤ k < j and (zk , zk+1 ) ∈ E(ΓG,(x,y) ) for all odd 0 < k < j}. The following result was obtained in [14]. Theorem 1.2 Let G be a finite group, and let M1 and M2 be distinct conjugate maximal subgroups of G. Then the subgroups (M1 , M2 )6 and (M2 , M1 )6 coincide and are normal in the group G. Under the hypothesis of Theorem 1.1, if M1 := Gx and M2 := Gy , where x and y [i] [i] are adjacent vertices of the graph Γ, then Gx ≤ (M1 , M2 )i and Gy ≤ (M2 , M1 )i for all i ∈ N. Thus, Theorem 1.1 follows from Theorem 1.2, and hence Theorem 1.2 can also be considered as a result establishing the validity of a strengthened version of the Sims conjecture. The following result is also derived from Theorem 1.2 (see [15, Corollary]). Corollary 1.3 Let G be a finite group, M1 be a maximal subgroup of G, and M2 be a subgroup of G containing (M1 )G and not contained in M1 . Then the subgroup (M1 , M2 )12 coincides with (M1 )G . The following examples show that the constant 6 in Theorems 1.1 and 1.2 and the constant 12 in Corollary 1.3 can not be decreased. (In what follows, we use standard terminology and notation concerning groups, see, for example, [1, 7, 5].) Example 1.4 Let G = E8 (q), where q is a power of a prime p, let M1 be a parabolic maximal subgroup of G obtained from the Dynkin diagram for E8 by deleting the root α4 , and let a be an element of the monomial subgroup of G corresponding to

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the reflection sα4 (see [3]). Define Q = Op (M1 ) and M2 = aM1 a−1 . Let Γ be the graph with vertex set {hM1 h−1 | h ∈ G} and edge set {{hM1 h−1 , hM2 h−1 } | h ∈ G}. Then, Γ and G (acting by conjugation) satisfy the hypothesis of Theorem 1.1. It can be shown that the series 1 = (M1 , M2 )6 < (M1 , M2 )5 < (M1 , M2 )4 < (M1 , M2 )3 < (M1 , M2 )2 < Op ((M1 , M2 )1 ) < Q coincides with the series [5] [4] [3] [2] [1] 1 = G[6] x < Gx < Gx < Gx < Gx < Op (Gx ) < Q,

where x = M1 ∈ V (Γ), and coincides with the upper and lower central series of the group Q. Example 1.5 Take G, M1 , and a as in Example 1.4. Define M2 = (M1 ∩ aM1 a−1 )a . Then, using properties indicated in Example 1.4, it is easy to verify that (M1 , M2 )11 = 1 (and (M1 , M2 )12 = 1). Note also that in Corollary 1.3, the assumption that M1 is a maximal subgroup in G is essential. In fact, as [15, Example 3] shows, there exists a sequence of triples (Gn , M1,n , M2,n ), n ∈ N, such that Gn is a finite group, M1,n and M2,n are non-maximal subgroups in Gn generating Gn , and (M1,n , M2,n )n = (M2,n , M1,n )n .

2

Yet more strong version of the Sims conjecture

Problem 2.1 formulated below can be considered as a yet more strong version of the Sims conjecture. Theorem 1.2 follows from its solution. We started a series of papers aimed at solution of Problem 2.1. Our obtained so far results are given in this section. Let G, M1 , and M2 satisfy the hypothesis of Theorem 1.2. We are interested in the case when (M1 )G = (M2 )G = 1 and 1 < |(M1 , M2 )2 | ≤ |(M2 , M1 )2 |. The set of all such triples (G, M1 , M2 ) is denoted by Π. We consider triples from Π up to the following equivalence: the triples (G, M1 , M2 ) and (G , M1 , M2 ) from Π are equivalent if there exists an isomorphism of G to G mapping M1 to M1 and M2 to M2 . The group G acts, by conjugation, faithfully and primitively on the set X = {gM1 g −1 | g ∈ G}. According to a refinement of the Thompson–Wielandt theorem for the case under consideration obtained by A. Fomin [9, Proposition 1] (see also [15, Proposition 1.1]) the product (M1 , M2 )2 (M2 , M1 )2 is a nontrivial p-group for some prime p. Depending on the form of the socle Soc(G) of the group G, we partition the set Π into the following subsets:

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Π0 is the set of triples (G, M1 , M2 ) from Π such that Soc(G) is not a simple non-abelian group, i.e., G is not an almost simple group; Π1 is the set of triples (G, M1 , M2 ) from Π with Soc(G) isomorphic to an alternating group; Π2 is the set of triples (G, M1 , M2 ) from Π \ Π1 with Soc(G) isomorphic to a simple group of Lie type over a field of a characteristic different from p; Π3 is the set of triples (G, M1 , M2 ) from Π \ (Π1 ∪ Π2 ) with simple Soc(G) isomorphic to a simple group of Lie type over a field of characteristic p; Π4 is the set of triples (G, M1 , M2 ) from Π with Soc(G) isomorphic to one of the 26 finite simple sporadic groups. For a nonempty set Σ ⊆ Π, define c(Σ) to be the maximal positive integer c such that (M1 , M2 )c−1 = 1 or (M2 , M1 )c−1 = 1 for some triple (G, M1 , M2 ) ∈ Σ, if such maximal number exists. Otherwise, we set c(Σ) = ∞. Define c(G, M1 , M2 ) = c({(G, M1 , M2 )}) and c(∅) = 0. It was announced in [14] that c(Π0 ) ≤ max1≤i≤4 c(Πi ), c(Π1 ) = 0, c(Π2 ) = 3, c(Π3 ) = 6, and c(Π4 ) = 5. Theorem 1.2 follows from the equality c(Π) = 6. Now we formulate a problem which, to be solved, implies Theorem 1.2 and can be considered as a yet more strong version of the Sims conjecture. Problem 2.1 Describe the set Π more precisely and find all triples from Π \ Π0 up to equivalence. The problem is of interest for the finite group theory. It concerns intersections of local maximal subgroups in finite almost simple groups. Although local maximal subgroups of finite almost simple groups are now classified, their intersections are not sufficiently investigated. If G is a finite almost simple group and (G, M1 , M2 ) ∈ Π\Π0 , then M1 and M2 are distinct conjugate local maximal subgroups in G whose intersection M1 ∩ M2 is, in a sense, large. The problem is also of interest for the graph theory since the set Π can be used to get a description of (undirected) connected finite graphs Γ whose automorphism [2] group G acts primitively on V (Γ) and Gx = 1 for x ∈ V (Γ). For any (G, M1 , M2 ) ∈ Π, let ΓG,{M1 ,M2 } be the graph defined by V (ΓG,{M1 ,M2 } ) = {gM1 g −1 | g ∈ G}, E(ΓG,{M1 ,M2 } ) = {{gM1 g −1 , gM2 g −1 } | g ∈ G}. Let Γ be a connected finite graph and G a primitive group of automorphisms of Γ. [2] Assume that Gx = 1 where x ∈ V (Γ). Identify vertices of Γ with their stabilizers in G. Then E(Γ) is the union of edge sets of some graphs of the form ΓG,{M1 ,M2 } , where M1 = Gx , M2 = Gy for {x, y} ∈ E(Γ) and (G, M1 , M2 ) ∈ Π (because ΓG,{M1 ,M2 } is a subgraph of Γ with V (ΓG,{M1 ,M2 } ) = V (Γ) and consequently the ball of radius 2 of Γ centered at x contains the ball of radius 2 of ΓG,{M1 ,M2 } centered at x). In particular, assuming in addition that G is edge-transitive, we get that Γ coincides with ΓG,{M1 ,M2 } , where M1 = Gx , M2 = Gy for {x, y} ∈ E(Γ) and (G, M1 , M2 ) ∈ Π.

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We started a series of papers aimed at solution of Problem 2.1. In the first paper [15] of this series, we proved the following two theorems. Theorem 2.2 Let (G, M1 , M2 ) ∈ Π0 . Then Soc(G) = T1 × · · · × Tk , k > 1, is a minimal normal subgroup of G with isomorphic simple non-abelian T1 , . . . , Tk , 1 = M1 ∩ Ti < Ti for each 1 ≤ i ≤ k, and M2 = gM1 g −1 for some g ∈ Soc(G). Furthermore, for 1 ≤ i ≤ k, let Hi := NG (Ti ), Hi := Hi /CG (Ti ), Hi,1 := NM1 (Ti )CG (Ti ) and Hi,2 := NM2 (Ti )CG (Ti ). Then Soc(Hi ) ∼ = Ti , Hi,1 and Hi,2 are maximal subgroups of Hi such that (Hi,1 )Hi = (Hi,2 )Hi = 1, and Hi,2 = gHi,1 g −1 . Finally, if 1 ≤ j ≤ k and M1 ∩0Tj = M2 ∩ Tj (at least one such j exists), then Hj,1 = Hj,2 , (Hj , Hj,k , Hj,3−k ) ∈ 1≤s≤4 Πs for some k ∈ {1, 2}, and c(G, M1 , M2 ) ≤ c(Hj , Hj,k , Hj,3−k ). As a consecuence, we have c(Π0 ) ≤ max1≤s≤4 c(Πs ). Theorem 2.3 The set Π1 is empty and, consequently, c(Π1 ) = 0. Theorem 2.2 follows from Theorem 1.1 of [15] in whose proof the O’Nan–Scott theorem (see [19]) is used. In the second paper [16] of the series, we proved the following theorem. Theorem 2.4 Let (G, M1 , M2 ) ∈ Π2 and Soc(G) be a simple group of exceptional Lie type. Then (M1 , M2 )3 = (M2 , M1 )3 = 1 and one of the following holds: (a) G ∼ = E ε (r) or G ∼ = E ε (r) : 2, ε ∈ {+, −}, r ≥ 5 is a prime, 9 | (r − 6

6

ε1), (M1 , M2 )2 = Z(O3 (M1 )) and (M2 , M1 )2 = Z(O3 (M2 )) are elementary abelian groups of order 33 , (M1 , M2 )1 = O3 (M1 ) and (M2 , M1 )1 = O3 (M2 ) are special groups of order 36 , the group M1 /O3 (M1 ) is isomorphic to SL3 (3) for G ∼ = E6ε (r) and is isomorphic to GL3 (3) for G ∼ = E6ε (r) : 2, the group M1 /O3 (M1 ) acts faithfully on O3 (M1 )/Z(O3 (M1 )) and induces the group SL3 (3) on Z(O3 (M1 )), |Z(O3 (M1 )) ∩ Z(O3 (M2 ))| = 32 and M1 ∩ M2 = NM1 ∩Soc(G) (Z(O3 (M1 )) ∩ Z(O3 (M2 ))); (b) G ∼ = Aut(3 D4 (2)), (M1 , M2 )2 = Z(M1 ) and (M2 , M1 )2 = Z(M2 ) are groups of order 3, not contained in Soc(G), M1 ∼ = Z3 × ((Z3 × Z3 ) : SL2 (3)), (M1 , M2 )1 = O3 (M1 ), (M2 , M1 )1 = O3 (M2 ) and M1 ∩ M2 is a Sylow 3subgroup in M1 .

In any case of items (a) and (b), the triples (G, M1 , M2 ) from Π exist and form one class up to equivalence. In Theorem 2.4, E6ε (r) is E6 (r) for ε = + and 2 E6 (r) for ε = −. In the proof of Theorem 2.4, we used the classification of local non-parabolic maximal subgroups in finite groups of exceptional Lie type [6, 20]. In the third paper [17] of the series, we proved the following theorem. Theorem 2.5 Let (G, M1 , M2 ) ∈ Π2 and Soc(G) be a simple group of classical non-orthogonal Lie type. Then (M1 , M2 )3 = (M2 , M1 )3 = 1 and one of the following holds:

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425

(a) G ∼ = Aut(L3 (3)), (M1 , M2 )2 = Z(M1 ) and (M2 , M1 )2 = Z((M2 ) are groups of order 2, not contained in Soc(G), M1 ∼ = Z2 × S4 , (M1 , M2 )1 = O2 (M1 ), 1 (M2 , M1 ) = O2 (M2 ) and M1 ∩ M2 is a Sylow 2-subgroup in M1 ; (b) G ∼ = U3 (8) : 31 or U3 (8) : 6, (M1 , M2 )2 = Z(M1 ), (M2 , M1 )2 = Z(M2 ) are groups of order 3, not contained in Soc(G), M1 ∼ = Z3 × (Z2 : SL2 (3)) 3

or Z3 × (Z23 : GL2 (3)), respectively, (M1 , M2 )1 = O3 (M1 ), (M2 , M1 )1 = O3 (M2 ) and M1 ∩ M2 is a Sylow 3-subgroup in M1 or its normalizer in M1 , respectively; (c) G ∼ = L4 (3) : 22 or Aut(L4 (3)), (M1 , M2 )2 = Z(M1 ), (M2 , M1 )2 = Z(M2 ) are groups of order 2, not contained in Soc(G), M1 ∼ = Z2 × S4 × S4 or Z2 × (S4  Z2 ), respectively, (M1 , M2 )1 = O2 (M1 ), (M2 , M1 )1 = O2 (M2 ) and M1 ∩ M2 is a Sylow 2-subgroup in M1 .

In any case of items (a), (b) and (c), the triples (G, M1 , M2 ) from Π exist and form one class up to equivalence. The description of Π2 was completed in the fourth paper [18] of the series, where the following theorem was proved. Theorem 2.6 Let (G, M1 , M2 ) ∈ Π2 and Soc(G) be a simple orthogonal group of 3 dimension ≥ 7. Then Soc(G) ∼ = P Ω+ 8 (r), where r is an odd prime, (M1 , M2 ) = (M2 , M1 )3 = 1 and one of the following holds: (a) r ≡ ±1 (mod 8), G contains an element inducing on Soc(G) a graph automorphism of order 3, (M1 , M2 )2 = Z(O2 (M1 )) and (M2 , M1 )2 = Z(O2 (M2 )) are elementary abelian groups of order 23 , (M1 , M2 )1 = O2 (M1 ) and (M2 , M1 )1 = O2 (M2 ) are special groups of order 29 , the group M1 /O2 (M1 ) is isomorphic to P SL3 (2) × Z3 or P SL3 (2) × S3 , and M1 ∩ M2 is a Sylow 2-subgroup in M1 ; (b) r ≤ 5, G/L either contains Outdiag(L) or is isomorphic to Z4 , (M1 , M2 )2 = Z(O2 (M1 ∩ L)) and (M2 , M1 )2 = Z(O2 (M2 ∩ L)) are elementary abelian groups of order 22 , (M1 , M2 )1 = [O2 (M1 ∩ L), O2 (M1 ∩ L)] and (M2 , M1 )1 = [O2 (M2 ∩ L), O2 (M2 ∩ L)] are elementary abelian groups of order 25 , O2 (M1 ∩ L)/[O2 (M1 ∩ L), O2 (M1 ∩ L)] is an elementary abelian group of order 26 , the group (M1 ∩ L)/O2 (M1 ∩ L) is isomorphic to the group S3 , |M1 : M1 ∩ M2 | = 24, and |M1 ∩ M2 ∩ L| = 211 . In any case of items (a) and (b), the triples (G, M1 , M2 ) from Π exist and form one class up to equivalence. In the proof of Theorems 2.5 and 2.6, we used the Aschbacher theorem on maximal subgroups of finite classical groups (see [12, 2]). Remark that we used GAP [10] to get some of the properties indicated in the item (b) of Theorem 2.6. We thank V. V. Korableva for help in the GAP calculations. We are planning to consider sets Π3 and Π4 in subsequent papers of the series. References [1] M. Aschbacher, Finite group theory, Cambridge University Press, Cambridge, 1986.

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[2] J. N. Bray, D. F. Holt and C. M. Roney-Dougal, The maximal subgroups of the lowdimensional finite classical groups, London Math. Soc. Lecture Note Ser., 407. Cambridge University Press, Cambridge, 2013. [3] N. Bourbaki, Groupes et alg`ebres de Lie, Hermann, Paris, 1968. [4] P. J. Cameron, C. E. Praeger, J. Saxl and G. M. Seitz, On the Sims conjecture and distance transitive graphs, Bull. London Math. Soc. 15 (1983), 499–506. [5] R. W. Carter, Simple groups of Lie type, John Wiley and Sons, London-New YorkSydney, 1972. [6] A. M. Cohen, M. W. Liebeck, J. Saxl and G. M. Seitz, The local maximal subgroups of exceptional groups of Lie type, finite and algebraic, Proc. London Mat. Soc. (3) 64 (1992), 21–48. [7] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford, 1985). [8] A. N. Fomin, On a conjecture of Sims, In Sixth all-Union Symp. on group theory, Naukova Dumka, Kiev, 1980, 158–165 [in Russian]. [9] A. N. Fomin, Properties of suborbits of finite primitive permutation groups, In Research in Group Theory, Izd. UrO RAN, Sverdlovsk, 1990, 87–94 [in Russian]. [10] The GAP Group, GAP — Groups, Algorithms, and Programming, Ver. 4.9.1. 2018. URL: http://www.gap-system.org. [11] P. B. Kleidman, The maximal subgroups of the finite 8-dimensional orthogonal groups P Ω+ 8 (q) and of their automorphism groups, J. Algebra 110 (1987), 173–242. [12] P. B. Kleidman and M. W. Liebeck, The subgroup structure of the finite classical groups, London Math. Soc. Lecture Note Ser., 129, Cambridge University Press, Cambridge, 1990. [13] W. Knapp, On the point stabilizer in a primitive permutation group, Math. Z. 133 (1973), 137–168. [14] A. S. Kondrat’ev and V. I. Trofimov, Stabilizers of graph’s vertices and a strengthened version of the Sims conjecture, Dokl. Math. 59 (1999), 113–115. [15] A. S. Kondrat’ev and V. I. Trofimov, Stabilizers of vertices of graphs with primitive automorphism groups and a strong version of the Sims conjecture. I, Proc. Steklov Inst. Math. 289 (2015), Suppl. 1, S146–S155. [16] A. S. Kondrat’ev and V. I. Trofimov, Stabilizers of vertices of graphs with primitive automorphism groups and a strong version of the Sims conjecture. II, Proc. Steklov Inst. Math. 295 (2016), Suppl. 1, S89–S100. [17] A. S. Kondrat’ev and V. I. Trofimov, Stabilizers of vertices of graphs with primitive automorphism groups and a strong version of the Sims conjecture. III, Proc. Steklov Inst. Math. 299 (2017), Suppl. 1, S113–S122. [18] A. S. Kondrat’ev and V. I. Trofimov, Stabilizers of vertices of graphs with primitive automorphism groups and a strong version of the Sims conjecture. IV, Trudy Inst. Mat. Mekh. UrO RAN 24 (2018), no. 3, 109–132 [in Russian]; English translation in Proc. Steklov Inst. Math., Suppl., to appear. [19] M. W. Liebeck, C. E. Praeger and J. Saxl, On the O’Nan–Scott theorem for finite primitive permutation groups, J. Austral. Math. Soc. Ser. A 44 (1988), 389–396. [20] M. W. Liebeck, J. Saxl and G. M. Seitz, Subgroups of maximal rank in finite exceptional groups of Lie type, Proc. London Math. Soc. (3) 65 (1992), 297–325. [21] C. Sims, Graphs and finite permutation groups, Math. Z. 95 (1967), 76–86. [22] J. G. Thompson, Bounds for orders of maximal subgroups, J. Algebra 14 (1970), 135– 138. [23] W. T. Tutte, A family of cubical graphs, Proc. Camb. Phil. Soc. 43 (1947), 459–474. [24] W. T. Tutte, On the symmetry of cubical graphs, Canad. J. Math. 11 (1959), 621–624. [25] H. Wielandt, Subnormal subgroups and permutation groups, Ohio State University Lecture Notes, Columbus, 1971.

ON THE CHARACTER DEGREES OF A SYLOW p-SUBGROUP OF A FINITE CHEVALLEY GROUP G(pf ) OVER A BAD PRIME TUNG LE∗ , KAY MAGAARD1 and ALESSANDRO PAOLINI§ ∗

Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa Email: [email protected]

§

FB Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany Email: [email protected]

Abstract Let q be a power of a prime p and let U (q) be a Sylow p-subgroup of a finite Chevalley group G(q) defined over the field with q elements. We first give a parametrization of the set Irr(U (q)) of irreducible characters of U (q) when G(q) is of type G2 . This is uniform for primes p ≥ 5, while the bad primes p = 2 and p = 3 have to be considered separately. We then use this result and the contribution of several authors to show a general result, namely that if G(q) is any finite Chevalley group with p a bad prime, then there exists a character χ ∈ Irr(U (q)) such that χ(1) = q n /p for some n ∈ Z≥0 . In particular, for each G(q) and every bad prime p, we construct a family of characters of such degree as inflation followed by an induction of linear characters of an abelian subquotient V (q) of U (q).

1

Introduction

Let q be a power of a prime p, and let Fq denote the field with q elements. A major research problem in the representation theory of finite groups is to understand the characters of a finite Chevalley group G(q) defined over Fq . Namely finite Chevalley groups contribute to a large part of all finite nonabelian simple groups. The study of the set Irr(G(q)) of ordinary irreducible characters of G(q) has been carried out to an extensive progress, starting from the groundbreaking work of Deligne and Lusztig [3], to developments which allowed to compute and process the character table of G(q) in [2] when the rank of G(q) is small. In particular, the set cd(G(q)) = {χ(1) | χ ∈ Irr(G(q))} of irreducible character degrees of G(q) is essentially known. The situation is different when we consider characters of G(q) over an algebraically closed field of characteristic = p, the so called cross-characteristics case. One has considerably less amount of information for such characters. The problem of studying such characters is closely related to the one of parametrizing the ordinary irreducible characters of a fixed Sylow p-subgroup U (q) of G(q), which we also denote by U (G(q)). Namely the induction of ψ ∈ Irr(U (q)) to G(q) remains an 1

The second author passed away on July 26th, 2018.

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-projective character, as and p are different. A decomposition of such induced character can be provided, if the behavior of ψ is known and we have enough information about the fusion of the conjugacy classes of U (q) to G(q). Even for groups of small rank, the set cd(U (q)) is more complicated to describe than the set cd(G(q)), and is in general also much bigger. We summarize here some of the main known results in this direction. If G(q) = An−1 (q) for n ≥ 2, then every degree of a character in Irr(U (q)) is a power of q [12], and in fact cd(U (q)) = {1, q, . . . , q μ(n) }, where μ(n) = m2 − m if n = 2m and μ(n) = m2 if n = 2m + 1, see [10] and [13]. If G(q) = Dn (q) for n ≥ 4 and p is odd, then cd(U (q)) = {1, q, . . . , q f (n) } with f (n) = n(n − 1)/2 − [n/2] [18]. A similar result holds for other G(q) of classical type for odd p; in particular, cd(U (q)) ⊆ {q n | n ∈ Z≥0 } for G(q) classical if and only if p is an odd prime, see [20] and [19]. Via the Kirillov orbit method, it was proved in [6] that if p is at least the Coxeter number of G(q) then cd(U (q)) ⊆ {q n | n ∈ Z≥0 } for G(q) an arbitrary finite Chevalley group. When p is a bad prime for G(q), the set cd(U (q)) is often not known. The goal of this work is to show that cd(U (q)) ⊆ {q n | n ∈ Z≥0 } does never occur in this case, by means of an explicit construction of a character of degree q n /p for some positive integer n. We first determine a parametrization of Irr(U (q)) when G(q) is of type G2 . We provide full details just for the primes p = 2 and p = 3; the result is straightforward from [4, Algorithm 3.3] if p ≥ 5. We use the subsequent Lemma 2.1 to parametrize certain characters of Irr(U (q)), and a counting argument to see that these determine all of Irr(U (q)). In particular, we find characters of degree q/p for p ∈ {2, 3}. Theorem 1.1 Let G(q) = G2 (3f ) or G(q) = G2 (2f ). The irreducible characters of U (q) are parametrized in Table 2. In particular, we have that ⎧ 3 2 ⎪ ⎨q + 2q − q − 1, 3 | Irr(U (q))| = 2q + 5q 2 − 10q + 4, ⎪ ⎩ 3 q + 5q 2 − 7q + 2,

if p ≥ 5, if p = 3, if p = 2.

It is not difficult to produce characters of degree q/2 in U (B2 (q)) when q is a power of 2, which we can inflate to U (Bn (q)), U (Cn (q)) and U (F4 (q)). We could similarly inflate characters of U (D4 (2f )) of degree q 3 /2, obtained in [8], to U (Dn (2f )) and U (Ek (2f )), for k = 6, 7, 8. For q = 3f , we have characters of degree q 4 /3 in type F4 (q) [4] and of degree q 7 /3 in type E6 [14] which we can inflate to U (Ek (2f )) for k = 7, 8. The work [14] also gives an example of an irreducible character of U (E8 (5f )) of degree q 16 /5. Finally, the construction of characters of degrees q/2 in Irr(U (G2 (2f ))) and q/3 in Irr(U (G2 (3f ))) follows from Theorem 1.1. This collection of results allows us to state the following. Theorem 1.2 Let G(q) be a finite Chevalley group over Fq . If p is a bad prime for G(q), then there exist χ ∈ Irr(U (q)) and some n ≥ 1 such that χ(1) = q n /p. In particular, families of characters of such degree are constructed as an inflation,

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429

followed by an induction of a linear character of an abelian subquotient V (q) of U (q), with labels as in Table 1. The labels of the characters are given as in [4]. In general, a label of the form ai (respectively bj ) of χ ∈ Irr(U (q)) corresponds to an element of F× q (respectively Fq ), which is the value on xi (1) (respectively xj (1)) of the linear character that we inflate and induce to obtain χ. More details on these labels are given in the sequel for each case taken into consideration. The importance of the construction of such characters lies in the fact that these could replace some classes of characters, defined just for good p, helpful for investigating the cross-characteristics representations of G(q). Let us for instance take G(q) = D4 (q), and = p with | q + 1. The decomposition numbers are obtained in [7] in the case when p is an odd prime; this assumption is required for exploiting properties of the generalized Gelfand-Graev characters and the parametrization of Green functions in [16]. A calculation shows that in the case p = 2, by inducing to G(q) the four irreducible characters of U (q) corresponding to (d1,2,4 , d3 ) ∈ F2 × F2 in Proposition 4.3 we get characters that play the role of the -projective characters Φ6 , . . . , Φ9 in [7, §5], which turn out to be of major importance to determine the unitriangular shape of the decomposition matrix of D4 (2f ). We now examine the question of whether each of the families of irreducible characters of degree q n /p in Table 1 consists of all the characters of Irr(U (q)) of such degree. It turns out that the previously mentioned works also determine that if G(q) is not E8 (5f ), then the families in Table 1 that do not arise from an embedding of a root system of smaller rank give in fact all the characters of Irr(U (q)) of degree q n /p for the corresponding values of n. On the other hand, if a family F does arise from an embedding of a smaller root system, then it is straightforward to get other characters in Irr(U (q)) of the same degree which are not in F by tensoring the characters in F with linear characters of a certain root subgroup indexed by a root in the difference of the two root systems. We then propose the following conjecture. Conjecture 1.3 The family in Table 1 of irreducible characters of degree q 16 /5 in Irr(U (E8 (5f ))) consists of all irreducible characters of U (E8 (5f )) whose degree is not a power of q. Finally, we present further progress and a question on fractional degrees in U (q). The work [5] provides a construction of irreducible characters of fractional degrees with denominator of the form pt with t ≥ 2. Namely if G(q) = Cn (q) then there exist irreducible characters of U (q) of degree q n(n−1)/2 /2t for every 0 ≤ t ≤ [n/2], and if G(q) = Dn then there exist irreducible characters of U (q) of degree q 3r(r−1)/2 /2t (respectively q 3r(r+1)/2 /2t ) for every 1 ≤ t ≤ [r/2] if n = 2r (respectively if n = 2r + 1). On the one hand, such characters seem to maximize the power m for character degrees of the form q m /pt . On the other hand, they seem not to maximize t, as in the case of U (D6 (2f )) one just gets t = 1 by applying the above formula, while by [15] we know that there also exist characters of the form q m /2t with t = 2 in U (D6 (2f )). It would be interesting to determine, in general,

Le, Magaard, Paolini: Character degrees of a Sylow p-subgroup of G(pf )

430 Type Bn

Bad p p=2

Character labels a2n−1 ,a3n−2 χdn−1 ,dn

Size of family 4(q − 1)2

Degree q/2

Unique iff n = 2

Cn

p=2

a ,a3n−2 χdn2n−1 ,dn−1 an+1,n+2,n+3 ,a2n ,a2n+1 ,a2n+2 χd1,2,4 ,d3 ,a4 χad13 ,d 2 ∗ a ,a χe13,e25 ,a9 χad26 ,d 3 a11 ,a12 ,a13 ,a∗6 χe1,3,4,7 ,e2 a8,9,10 ,a13 ,a14 ,a15 χd2,3,5 ,d4 a17 ,a18 ,a19 ,a20 ,a21 ,a∗8,9,10 χe2,1,3,5,6,7,11 ,e4 a9,10,11 ,a15 ,a16 ,a17 χd2,3,5 ,d4 a20 ,a21 ,a22 ,a23 ,a24 ,a∗9,10,11 χe2,1,3,5,6,8,12 ,e4 a10,11,12 ,a17 ,a18 ,a19 χd2,3,5 ,d4 a23 ,a24 ,a25 ,a26 ,a27 ,a∗10,11,12 χe2,1,3,5,6,9,13 ,e4 a37 ,...,a43 ,a∗12,13 χf1...,4,6,...,11,14,...,17,22,23 ,f5

4(q − 1)2

q/2

iff n = 2

4(q −

1)4

q 3 /2

iff n = 4

4(q − 1)2

q/2

yes

1)2 /2

9(q − 4(q − 1)2

q/3 q/2

yes no

9(q − 1)4 /2 4(q − 1)4

q 4 /3 q 3 /2

yes no

9(q − 1)6 /2 4(q − 1)4

q 7 /3 q 3 /2

yes no

9(q − 1)6 /2 4(q − 1)4

q 7 /3 q 3 /2

no no

9(q − 1)6 /2

q 7 /3

no

25(q −

q 16 /5

Conj. 1.3

Dn G2 F4 E6 E7

p=2 p=2 p=3 p=2 p=3 p=2 p=3 p=2 p=3 p=2

E8

p=3 p=5

1)8 /4

Table 1. Families of characters of U (q) of degree q n /p for some n ∈ N, and their uniqueness of such degree, for each bad prime p and every Lie type. all powers q m /pt that can occur as character degrees for Irr(U (q)), in particular the maximum value of t, for each finite Chevalley group G(q).

2

Preliminaries

We first let G be any finite group, H be a subgroup of G, and N be a normal subgroup of G. We recall some notation on characters of G and its subgroups. We let Irr(G) be the set of irreducible characters of the group G. For a character χ ∈ Irr(G), we denote by ker(χ) the kernel of χ and by Z(χ) its centre. We denote by χ|H the restriction of χ to H. Let ϕ ∈ Irr(G/N ). Then we denote by Inf G G/N ϕ ∈ Irr(G) the inflation of the character ϕ to G. If ψ is a character of H, then we denote by IndG H (ψ) the induction of the character ψ to G. We denote by  , the usual inner product defined on the characters of G. If η ∈ Irr(H), then we denote Irr(G | η) = {χ ∈ Irr(G) | χ|H , η =  0} = {χ ∈ Irr(G) | χ, IndG  0}. H η = We recall a result that we use several times in the sequel. The proof is a particular case of [9, Lemma 2.1] detailed in [4, §4.1]. Lemma 2.1 Let G be a finite group, let H ≤ G and let X be a transversal of H in G. Let Y and Z be subgroups of H, and λ ∈ Irr(Z). Suppose that

Le, Magaard, Paolini: Character degrees of a Sylow p-subgroup of G(pf )

431

(i) Z ⊆ Z(G), (ii) X and Y are elementary abelian groups with |X| = |Y |, (iii) Y  H, (iv) Z ∩ Y = 1, and (v) the commutator group [X, Y ] is contained in Z. If we put

X  := {x ∈ X | λ([x, y]) = 1 for all y ∈ Y }

and

Y  := {y ∈ Y | λ([x, y]) = 1 for all x ∈ X},

and if Y˜ is a complement of Y  in Y , then the map 

HX  ˜ IndG HX  Inf HX  /Y˜ ker λ : Irr(HX /Y ker λ | λ) −→ Irr(G | λ)

(1)

is a bijection. We keep the notation for q, G(q), U (q) and Irr(U (q)) as in the Introduction. We briefly recall the notion of bad primes. Let Φ be the root system associated with G(q), and let Φ+ be set of positive roots in Φ. We fix an enumeration α1 , . . . , α|Φ+ | of the positive roots, with α1 , . . . , αn the simple roots of Φ+ , and α0 := α|Φ+ | the highest root in Φ+ . We say that p is a bad prime for Φ+ if p divides one of the coefficients of α0 in its linear combination in terms of simple roots. We recall that as G(q) is a split group, we have that  U (q) = Xα , with Xα = {xα (t) | t ∈ Fq } ∼ = (Fq , +), α∈Φ+

hence we have |U (q)| = q |Φ | . The group Xα is called the root subgroup of U (q) associated to α ∈ Φ+ , and each element xα (t) is called the root element with respect to α ∈ Φ+ and t ∈ Fq . We say P ⊆ Φ+ is a pattern in Φ+ if for every α, β ∈ P, either α + β ∈ P or α+β ∈ / Φ+ . For a pattern P, we have that the product  Xα XP := +

α∈P

is well defined, and it is a subgroup of U (q). We call XP the pattern group corresponding to P. If P := {αi1 , . . . , αim }, then we also write X{i1 ,...,im } for XP ; similarly we write xi (t) for the root element xαi (t), with αi ∈ Φ+ and t ∈ Fq . A subset N of a pattern P is normal in P, or N  P, if for every α ∈ N and β ∈ P, one has either α + β ∈ N or α + β ∈ / Φ+ . It is easy to check that if N  P, then XN  XP . For χ ∈ Irr(U (q)), we define the central root support rs(χ) := {α ∈ Φ+ | Xα ⊆ Z(χ) and Xα ker(χ)}. In order to construct the subquotients V (q) as in Theorem 1.2, and to parametrize the corresponding characters, we need to fix a nontrivial character of (Fq , +). Denote by Tr : Fq → Fp ∼ = Zp the trace map of the field extension Fq | Fp . We define φ : Fq → C× by φ(t) = e

i2π Tr(t) p

for t ∈ Fq . Notice that

ker φ = {tp − t | t ∈ Fq }.

(2)

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Remark 2.2 In the sequel, for each cyclic group C of order p we implicitly fix a morphism ϕ : Z/pZ → C, and for each d ∈ Z/pZ we denote by μdC ∈ Irr(C) the character such that μdC (ϕ(1)) = ζpd , where ζp is a fixed primitive p-th root of unity.

3

A parametrization of Irr(U (G2 (q))), when q = 2f or q = 3f

In this section we provide a parametrization of the irreducible characters of U (q) when G(q) = G2 (q) for every prime p. This is done by parametrizing families of characters of certain subquotients of U (q), and checking by using the well-known formula  |U (q)| = χ(1)2 (3) χ∈Irr(U (q))

that these families give in fact all of Irr(U (q)). We will denote by T1 , T2 , . . . such ai ,...,ai subquotients of Irr(U (q)) in the sequel. Each of the labels χbj 1 ,...,bj r in Tables 1 and s 1 2, with ai1 , . . . , air ∈ F× q and bj1 , . . . , bjs ∈ Fq , is obtained in a similar way as in [4], namely by inflation-induction process of the corresponding family of characters λai1 ,...,air ⊗ μj1 ⊗ · · · ⊗ μjs ∈ Irr(V (q)). The characters of Irr(U (G2 (q))) with labels of the form d ∈ F2 or e ∈ F3 are described in more detail in this section. We denote by α1 the long simple root in G2 , hence α2 is its short simple root. Recall that in this case we have Φ+ = {α1 , . . . , α6 }, where α3 = α1 + α2 ,

α4 = α1 + 2α2 ,

α5 = α1 + 3α2 ,

α6 = 2α1 + 3α2 .

For every prime p, and for every s, t ∈ Fq , the commutator relations among root elements are as follows, [x1 (s), x2 (t)] = x3 (−st)x4 (st2 )x5 (−st3 )x6 (−s2 t3 ), [x2 (s), x3 (t)] = x4 (2st)x5 (−3s2 t)x6 (3st2 ), [x1 (s), x5 (t)] = x6 (st),

[x2 (s), x4 (t)] = x5 (3st),

[x3 (s), x4 (t)] = x6 (−3st),

and [xi (s), xj (t)] = 1 in the remaining cases. Observe that the irreducible characters of U (G2 (q)) when p ≥ 5 can be easily parametrized by [4, Algorithm 3.3]. Proposition 3.1 Let q = pf with p ≥ 5. Then U (G2 (q)) has exactly (i) q(q − 1) irreducible characters of degree q 2 , (ii) q 2 (q − 1) + q(q − 1) + (q − 1) irreducible characters of degree q, and (iii) q 2 linear characters. The characters of degree q 2 are precisely the ones with central root support {α6 }, and each of the summands q i (q − 1) in the expression for the number of irreducible characters of degree q corresponds to the family of irreducible characters with central root support {αi+3 } for i = 0, 1, 2. We examine next the case q = 2f .

Le, Magaard, Paolini: Character degrees of a Sylow p-subgroup of G(pf )

433

Proposition 3.2 Let q = 2f and G(q) = G2 (q). Then U (q) has exactly (i) q(q − 1) irreducible characters of degree q 2 , (ii) (q − 1)q 2 + 2(q − 1) irreducible characters of degree q, (iii) 4(q − 1)2 irreducible characters of degree q/2, and (iv) q 2 linear characters. Proof Let T1 := U (q), and let Z := Z(T1 ) = X6 . Let us define λa6 ∈ Irr(Z) in the usual way. By the commutator relations, it is an easy check to deduce that the assumptions of Lemma 2.1 are verified with X := X1 X3 , Y := X4 X5 and H := X2 Y Z. We have that X  = Y  = 1. Let V1 (q) := X2 X6 . Then the family F1 := {IndV1 (q)Y Inf V11 (q) (λa6 ⊗ μ2 ) | a6 ∈ F× q , μ2 ∈ Irr(X2 )} U (q)

V (q)Y

consists of q(q − 1) irreducible characters of U (q) of degree q 2 . Let now T2 := U (q)/X6 . We have Z := Z(T2 ) = X5 . Again we apply Lemma 2.1; it is an easy check that its hypotheses are satisfied with X := X2 , Y := X4 and H := X1 X3 Y Z. We have X  = Y  = 1 also in this case. If V2 (q) := X1 X3 X5 , then we have that F2 := {IndV2 (q)Y Inf V22 (q) (λa5 ⊗ μ1 ⊗ μ3 ) | a5 ∈ F× q , μ1 ∈ Irr(X1 ), μ3 ∈ Irr(X3 )} U (q)

V (q)Y

is a family of q 2 (q − 1) irreducible characters of degree q. We now notice that T3 := U (q)/X5 X6 is isomorphic to U (B2 (2f )) in the obvious way. By the subsequent Proposition 4.1, we get a family F3 of 2(q − 1) irreducible characters of degree q, a family F4 of 4(q − 1)2 irreducible characters of degree q/2, and the family F5 of q 2 linear characters. Finally, notice that if χi is one of the characters in Fi , for i = 1, . . . , 5, then we have 5 

χi (1)2 |Fi | = q 5 (q − 1) + q 4 (q − 1) + 2q 2 (q − 1) + 4q 2 (q − 1)2 /4 + q 2

i=1

= q 6 = |U (q)|, hence by Equation (3) we have F1 ∪ · · · ∪ F5 = Irr(U (q)). We now determine the irreducible characters of U (G2 (q)) when q = 3f . Proposition 3.3 Let q = 3f and G(q) = G2 (q). Then U (q) has (i) (q − 1)2 irreducible characters of degree q 2 , (ii) 2(q − 1)q 2 + (q − 1)(q + 3)/2 irreducible characters of degree q, (iii) 9(q − 1)2 /2 irreducible characters of degree q/3, and (iv) q 2 linear characters.



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Proof Let T1 := U (q). We have that Z(T1 ) = X4 X6 . Let us put Z := X6 and let us define λa6 as usual for a6 ∈ F× q . Then Lemma 2.1 applies with X := X2 , Y := X5 and H := X1 X3 Y Z, and X  = Y  = 1. Let V1 (q) := X1 X3 X4 X6 . Then V1 (q) ∼ = (X1 X3 X4 ) × X6 = T  × Fq , where T  is a special group of the form q 1+2 with Z(T  ) = X4 . One then has that T  has q − 1 irreducible characters of degree q and q 2 linear characters. Hence we get two families of characters, namely X

F1 = {IndX{1,3,4,6} Inf X{1,3,4,6} (λa4 ,a6 ) | a4 , a6 ∈ F× q }, 4 X6 U (q)

which consists of (q − 1)2 characters of U (q) of degree q 2 , and X

(λa6 ⊗ μ1 ⊗ μ3 ) | a6 ∈ F× F2 = {IndX{1,3,4,5,6} Inf X{1,3,4,5,6} q , μ1 ∈ Irr(X1 ) {1,3,6} and μ3 ∈ Irr(X3 )}, U (q)

which has q 2 (q − 1) characters of U (q) of degree q. Let us now define T2 := U (q)/X6 . Then X4 ⊆ Z(T2 ). We let Z := X4 and λa4 ∈ Irr(Z). It is a straightforward check that X := X2 , Y := X3 and H := X1 X5 Y Z satisfy the assumptions of Lemma 2.1. Again we have that X  = Y  = 1. Notice that V2 (q) := X1 X4 X5 X6 /X6 is an abelian group. Hence we get a family X

(λa4 ⊗ μ1 ⊗ μ5 ) | a4 ∈ F× F3 = {IndX{1,3,4,5,6} Inf V2{1,3,4,5,6} q , μ1 ∈ Irr(X1 ) (q) and μ5 ∈ Irr(X5 )} U (q)

of q 2 (q − 1) irreducible characters of U (q) of degree q. We now let T3 := U (q)/X{4,6} . In this case, we have Z := Z(T3 ) = X{3,5} , and we define λa3 ,a5 ∈ Irr(Z) for a3 , a5 ∈ Fq in a similar way as in the case of T1 and T2 . The groups X := X1 , Y := X2 and H := X{2,3,5} satisfy the hypotheses of Lemma 2.1. We now want to compute the sets X  and Y  . We have that λ([x2 (t), x1 (s)]) = λ(x3 (st)x5 (st3 )) = φ(st(a3 + a5 t2 )). Let us first assume that a3 , a5 ∈ F× q , and that −a3 /a5 is a square. In this case, we write a∗3 for a3 . Notice that there are (q − 1)2 /2 such pairs of elements a∗3 , a5 × × in F× q . Namely the set S of squares in Fq is a subgroup of Fq of order (q − 1)/2, ∗ × ∗ and −a3 /a5 is a square if and only if a5 ∈ Fq and a3 ∈ −a5 S. Let ω3,5 be a fixed square root of −a∗3 /a5 . By Equation (1), we have that X  := {1, x1 (±1/(a3 ω3,5 ))}

Y  := {1, x2 (±ω3,5 )}.

and

In this case we have [G : HX  ] = q/3. Moreover, V3 (q) := ZX  Y  ∼ = HX  /Y˜ ker λ is abelian. By Lemma 2.1, we obtain a family a∗ ,a

∗ F4 := {χe13,e25 | e1 , e2 ∈ F3 , a5 ∈ F× q and a3 ∈ −a5 S},

where

a∗ ,a

U (q)



∗

Y a3 ,a5 ⊗ μeX1 ⊗ μeY2 ), χe13,e25 := IndHX  Inf HX V3 (q) (λ

of 9(q − 1)2 /2 irreducible characters of U (q) of degree q/3.

Le, Magaard, Paolini: Character degrees of a Sylow p-subgroup of G(pf ) Irr(U (G2 (2f ))) Labels Size of family χab26 q(q − 1) χab15,b3 q 2 (q − 1) χa3 ,0 q−1 χ0,a4 q−1 ,a4 χad13 ,d 4(q − 1)2 2 χb1 ,b2 q2

Irr(U (G2 (3f ))) Labels Size of family χa4 ,a6 (q − 1)2 χab16,b3 q 2 (q − 1) χab14,b5 q 2 (q − 1)  ,a a 5 χ 3 (q − 1)2 /2 a ,0 3 χ q−1 χ0,a5 q−1 a∗ ,a χe13,e25 9(q − 1)2 /2 χb1 ,b2 q2

Deg. q2 q q q q/2 1

435

Deg. q2 q q q q q q/3 1

Table 2. A parametrization of Irr(U (G2 (q))) for p = 2 and p = 3.  We now suppose that a3 , a5 ∈ F× q , and a5 /a3 is not a square. We write a3 for   ∼ a3 . In this case, we have that X = Y = 1. We put V4 (q) := H/Y = X3 X5 . We get a family 

a3 ,a5  × F5 := {IndHY Inf HY ) | a 5 ∈ F× q and a3 ∈ Fq \ −a5 S} V4 (q) (λ U (q)

of (q − 1)2 /2 irreducible characters of U (q) of degree q. If exactly one of a3 or a5 is nonzero, then we also get X  = Y  = 1. Let V5 (q) := X3 X5 . Then we get a family a3 ,a5 × F6 := {IndHY Inf HY ) | (a3 , a5 ) ∈ (F× q × {0}) ∪ ({0} × Fq )} V5 (q) (λ U (q)

of 2(q − 1) irreducible characters of degree q of U (q). The choice a3 = a5 = 0 corresponds to the family F7 of q 2 linear characters of U (q). Finally, if χi is any character in Fi , for i = 1, . . . , 7, then we have 7 

χi (1)2 |Fi | = q 4 (q − 1)2 + q 2 (2q 2 (q − 1) + (q − 1)(q + 3)/2) + q 2 (q − 1)2 /2 + q 2

i=1

= q 6 = |U (q)|, hence F1 ∪ · · · ∪ F7 = Irr(U (G2 (q))).

4



Characters of fractional degree of U (q) in classical type

We now focus on the characters of U (B2 (q)) when q = 2f . The family of characters of degree q/2 in U (B2 (q)) was obtained in [17, §7] and revisited in [1, §7] in the context of character sheaves. We construct it here as an inflation-induction process from some subquotient of U (B2 (q)). Proposition 4.1 There are exactly 4(q − 1)2 irreducible characters of U (B2 (q)) of degree q/2.

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Proof Since p = 2, we have that [x3 (s), x2 (t)] = 1. Hence Z := Z(U (q)) = X{3,4} . Let X := X1 , Y := X2 and H := X{2,3,4} . Then the assumptions of Lemma 2.1 a3 ,a4 ∈ Irr(Z) be such that are satisfied. Let us fix a3 , a4 ∈ F× q , and let λ = λ λ(xi (t)) = φ(ai t) for i = 3, 4. We have that λ([x1 (s), x2 (t)]) = λ(x3 (st)x4 (st2 )) = φ(st(a3 + a4 t)), and in the notation of Lemma 2.1, by Equation (2) we have X  := {1, x1 (a4 /a23 )}

and

Y  := {1, x2 (a3 /a4 )}.

Notice that [U (q) : HX  ] = q/2 and that HX  /Y˜ ker λ ∼ = ZX  Y  . Let us define V (q) := ZX  Y  . By Equation (1) we have that 

U (q)

IndHX  Inf HX V (q) : Irr(V (q) | λ) −→ Irr(G | λ) is a bijection. Moreover, we have that V (q) is abelian, hence ,a4 Irr(V (q) | λa3 ,a4 ) = {ψda13,d | d1 , d2 ∈ F2 }, 2

where ,a4 := λa3 ,a4 ⊗ μdX1 ⊗ μdY2 . ψda13,d 2

Notice that the sets Irr(G | λa3 ,a4 ) are disjoint for a3 , a4 ∈ F× q . Finally, recall by [17, §7] that the other characters in Irr(U (q)) consist of two families of size q − 1 of characters of degree q, namely the irreducible characters with central root support {α3 } and {α4 } respectively, and the family of the q 2 linear characters. 

α1

α2

α3

αn−1

αn

α1

α2

α3

αn−1

αn

Bn Cn

Figure 1. The Dynkin diagrams of Bn and Cn . Simple roots are labelled as in CHEVIE. By inflation of the irreducible characters in Proposition 4.1 we obtain the following. Corollary 4.2 Let G(q) = Bn (q) or G(q) = Cn (q), with n ≥ 3. Then U (q) has at least 4(q − 1)2 characters of degree q/2. Proof Let us define N :=

n−2  i=1

{α ∈ Φ+ | αi ≤ α}.

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437

We have that N  Φ+ , hence XN  U (q), and U (q)/XN ∼ = X{n−1,n,2n−1,3n−2} is isomorphic to U (B2 (q)), with αn−1 long and αn short in type Bn and viceversa in type Cn , in the notation of Figure 1. If a2n−1 , a3n−2 ∈ F× q and dn−1 , dn ∈ F2 , we then define U (q) a2n−1 ,a3n−2 a2n−1 ,a3n−2 χdn−1 := Inf U (q)/XN ψdn−1 ,dn ,dn if U (q) = U (Bn (q)), and a

U (q)

,a

a

,a

3n−2 3n−2 := Inf U (q)/XN ψdn2n−1 χdn2n−1 ,dn−1 ,dn−1

a

,a

a

,a

2n−1 3n−2 3n−2 if U (q) = U (Cn (q)), where ψdn−1 and ψdn2n−1 are defined as in Proposi,dn ,dn−1 tion 4.1. 

Let us now examine the groups of type D4 . The irreducible characters of U (D4 (q)) have been completely parametrized in [8] for every prime p. Unlike the case of type Bn and Cn , there are no characters of degree q/2 in type D4 . The even in statement and the construction below combine the study of the family F8,9,10 [8] and the approach of [4]. Proposition 4.3 Let G(q) = D4 (q). Then U (q) has exactly 4(q − 1)4 irreducible characters of degree q 3 /2. Construction. Let N := X{11,12} . Notice that N  Φ+ . For fixed a8 , a9 , a10 ∈ F× q , define x1,2,4 (t) := x1 (a10 t)x2 (a9 t)x4 (a8 t)

and

x5,6,7 (s) := x5 (a10 s)x6 (a9 s)x7 (a8 s)

for every s, t ∈ Fq . Let X  := {x1,2,4 (t) | t ∈ Fq }

and

Y  := {x5,6,7 (s) | s ∈ Fq },

and let Y˜ := X5 X6 . Then Y = Y  × Y˜ . a5,6,7 ,a8 ,a9 ,a10 by λ(x Let us also fix a5,6,7 ∈ F× 5,6,7 (s)) = q . We define λ = λ φ(a5,6,7 s), and λ(xi (ti )) = φ(ai ti ) for i = 8, 9, 10, and we define W1 := {1, x1,2,4 (a5,6,7 /(a8 a9 a10 )}

and

W2 := {1, x3 (a8 a9 a10 /a25,6,7 )}

and V (q) := W1 W2 Y Z/(Y˜ ker λ). Then each character of a

,a ,a9 ,a10

5,6,7 8 F := {ψd1,2,4 ,d3

| a5,6,7 , a8 , a9 , a10 ∈ F× q , d1,2,4 , d3 ∈ F2 },

where a

,a ,a9 ,a10

5,6,7 8 ψd1,2,4 ,d3

U (q)

X  W2 Y X{8,...,12}

:= IndX  W2 Y X{8,...,12} Inf V (q)

d

(λa5,6,7 ,a8 ,a9 ,a10 ⊗μW1,2,4 ⊗μdW32 ), 1 a

,a ,a ,a

5,6,7 8 9 10 are all distinct. is irreducible in U (q) of degree q 3 /2. The characters ψd1,2,4 ,d3 Finally, by [8], there are no other characters in Irr(U (q)) of degree q n /2 for any n ≥ 0. 

As done in the case of type Bn and Cn , we obtain by inflation characters of degree q 3 /2 in type Dn for every n ≥ 5.

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Corollary 4.4 For n ≥ 5, the group U (Dn (q)) has at least 4(q − 1)4 irreducible characters of degree q 3 /2. Proof In a similar way as in Corollary 4.2, we have that N :=

n 

{α ∈ Φ+ | αi ≤ α}  Φ+ .

i=5

Then we have that

U (q)/XN = XS ∼ = U (D4 (q)),

where S := {1, 2, 3, 4, n + 1, n + 2, n + 3, 2n, 2n + 1, 2n + 2, 3n − 1, 4n − 3}. We then apply Proposition 4.3, namely if an+1,n+2,n+3 , a2n , a2n+1 , a2n+2 ∈ F× q and d1,2,4 , d3 ∈ F2 , the characters as in the claim are given by a

n+1,n+2,n+3 χd1,3,4 ,d2

,a2n ,a2n+1 ,a2n+2 a

n+1,n+2,n+3 where each of the ψd1,3,4 ,d2

U (q)

a

n+1,n+2,n+3 := Inf U (q)/XN ψd1,3,4 ,d2

,a2n ,a2n+1 ,a2n+2

,a2n ,a2n+1 ,a2n+2

,

is defined as in Proposition 4.3.



Characters of fractional degree of U (q) in types F4 and Ek

5

We now move on to character degrees of the form q n /p in type F4 . We first consider the case of the prime p = 2. We have N := {α ∈ Φ+ | α1 ≤ α} ∪ {α ∈ Φ+ | α4 ≤ α}  Φ+ , and U (q)/XN ∼ = U (B2 (q)). Hence we obtain characters of degrees q/2 in U (F4 (q)). For p = 3, we have the following explicit construction. Proposition 5.1 ([4], §4.3) Let q = 3f , and let G(q) = F4 (q). There exist exactly 9(q − 1)4 /2 irreducible characters of U (q) of degree q 4 /3. Construction. Let N := {α14 , . . . , α24 }. Then N  Φ+ . For a11 , a12 , a13 ∈ F× q , let x1,3,4,7 (t) := x1 (a13 t)x1 (a13 t)x3 (a−12 t)x4 (a11 t)x7 (−a11 a12 t2 ) ∗ × for every t ∈ Fq . Moreover, let S be the set of squares in F× q , and for every a6 ∈ Fq ∗ ∗ such that a6 /a11 a12 a13 ∈ S let e be a square root of a6 /a11 a12 a13 , and let

X  := {x1,3,4,7 (t) | t ∈ Fq } and W1 := {x1,3,4,7 (es) | s ∈ F3 },

W2 := {x2 (t/(a11 a212 a13 e3 )) | t ∈ F3 }.

∗

Let λ := λa11 ,a12 ,a13 ,a6 be defined as λ(xi (t)) = φ(ai t) for i = 11, 12, 13 and λ(x6 (t)) = φ(a∗6 t). Let Y := X{5,8,9,10} , and let V (q) := W1 W2 X6 Y Z/ ker(λ). Then we have that a ,a ,a ,a∗6

11 12 13 F := {χe1,3,4,7 ,e2

∗ | a11 , a12 , a13 ∈ F× q , e1,3,4,7 , e2 ∈ F3 and a6 /a11 a12 a13 ∈ S},

Le, Magaard, Paolini: Character degrees of a Sylow p-subgroup of G(pf )

439

where a ,a ,a ,a∗6

11 12 13 χe1,3,4,7 ,e2



U (q)

∗

e

W2 X6 Y ZXN := IndX  W2 X6 Y ZXN Inf X (λa11 ,a12 ,a13 ,a6 ⊗ μW1,3,4,7 ⊗ μeW2 2 ), V (q) 1

is a family of 9(q − 1)4 /2 irreducible characters of U (q) of degree q 4 /3. By [4, Section 4], this family consists of all irreducible characters of U (q) of degree q n /3 for some n ≥ 0.  α2

α1

α3

α4

α5

α6

α7

α8

Figure 2. The Dynkin diagram of Ek for k = 6, 7, 8. Simple roots are labelled as in CHEVIE. We are left with the exceptional groups of types E6 , E7 and E8 . Let us first consider p = 2. We define Nk := {α ∈ Φ+ | α1 ≤ α} ∪

k 

{α ∈ Φ+ | αk ≤ α},

i=6

for k = 6, 7, 8. If Φ+ is a root system in type Ek , then Nk  Φ+ , and we have U (Ek (q))/XNk ∼ = U (D4 (q)). We obtain a family Fk of 4(q −1)4 characters of degree 3 q /2 of U (Ek (q)) by inflation from U (Ek (q))/XNk , for k = 6, 7, 8. We summarize in the following proposition the result obtained by the first and the second author in [14] for p = 3. Proposition 5.2 ([14], Section 3) Let q = 3f , and let G(q) = E6 (q). There exist exactly 9(q − 1)6 /2 irreducible characters of U (q) of degree q 7 /3. Construction. Let N = {α22 , . . . , α36 }. Notice that XN  U (q) and Z := Z(U (q)/XN ) ∼ = X{17,...,21} . For every t, r, s ∈ Fq we let x8,9,10 (t) := x8 (−t)x9 (t)x10 (t), x1,2,3,5,6,7,11 (t, r, s) := x2 (t)x1 (t)x3 (−t)x5 (t)x6 (−t)x7 (r)x11 (s), ∗ and for every square a∗8,9,10 in F× q and a fixed a square root e of a8,9,10 , let

X8,9,10 := {x8,9,10 (t) | t ∈ Fq } and F2 := {1, x2,1,3,5,6,7,11 (±e, 2e2 , 2e2 )},

F4 := {1, x4 (±1/e3 )}.

Le, Magaard, Paolini: Character degrees of a Sylow p-subgroup of G(pf )

440

Let V (q) := ZX8,9,10 F2 F4 . For e1 = e1,2,3,5,6,7,11 and e2 = e4 in F3 , we denote a∗

by λe18,9,10 ,e2 ∈ Irr(V (q)) the character such that a∗

a∗

∗ λe18,9,10 ,e2 (x8,9,10 (t)) = φ(a8,9,10 t),

a∗

e1 λe18,9,10 ,e2 |F2 = μF2 ,

e2 λe18,9,10 ,e2 |F4 = μF4 ,

and λ(xi (t)) = φ(t) for i = 17, . . . , 21, and we let H = ZX{8,9,10} X{12,...21} XN . Then we have that a∗

∗ × F := {χe18,9,10 ,e2 | e1 , e2 ∈ F3 and a8,9,10 is a square in Fq },

with

a∗

a∗

U (q)

8,9,10 HX4 F2 χe18,9,10 ,e2 := IndHX4 F2 Inf V (q) (λe1 ,e2 )

is a family of 9(q − 1)/2 irreducible characters of degree q 7 /3. A split maximal torus of G(q) acts transitively on Irr(X17 )× × · · · × Irr(X21 )× ; a17 ,...,a21 ,a∗8,9,10

here Irr(Xi )× denotes Irr(Xi ) \ {1Xi }. In particular, if λe1 ,e2 is defined such that a17 ,...,a21 ,a∗8,9,10

λe1 ,e2

∈ Irr(V (q))

a∗

|X8,9,10 F2 F4 = λe18,9,10 ,e2 |X8,9,10 F2 F4

and λ(xi (t)) = φ(ai t) for i = 17, . . . , 21, then we obtain the family a17 ,...,a21 ,a∗8,9,10

F  = {χe1 ,e2 where

∗ × | e1 , e2 ∈ F3 , a17 , . . . , a21 ∈ F× q , a8,9,10 is a square in Fq },

a17 ,...,a21 ,a∗8,9,10

χe1 ,e2

U (q)

a17 ,...,a21 ,a∗8,9,10

4 F2 := IndHX4 F2 Inf HX V (q) (λe1 ,e2

),

which consists of 9(q − 1)6 /2 elements of Irr(U (q)) of degree q 7 /3. Finally, by [15] the family F  consists of all irreducible characters of U (q) of degree q n /3 for some n ≥ 0.  The construction in Proposition 5.2 allows us to produce characters of degree q 7 /3 also in U (E7 (3f )) and U (E8 (3f )). Let k ∈ {7, 8}, and let Nk :=

k 

{α ∈ Φ+ | αk ≤ α}.

i=7

Then Nk  Φ+ , and we inflate the family of 9(q − 1)6 /2 irreducible characters of U (Ek (q))/XNk ∼ = U (E6 (q)) obtained in Proposition 5.2 to U (Ek (q)). We finish by constructing irreducible characters of degree q 16 /5 in U (E8 (5f )). Proposition 5.3 ([14], Section 4) Let q = 5f , and let G(q) = E8 (q). Then there exist at least 25(q − 1)8 /4 irreducible characters of U (q) of degree q 16 /5. Construction. The set N := {α44 , . . . , α120 } is a normal subset of Φ+ , and Z := ∗ 4 × U (q)/XN ∼ = X{37,...43} . Fix a∗37,...,43 ∈ F× q such that a37,...,43 = e for some e ∈ Fq .

Le, Magaard, Paolini: Character degrees of a Sylow p-subgroup of G(pf )

441

Observe that such an element a∗37,...,43 ∈ F× q can take (q − 1)/4 distinct values in F× q . For every u1 , u2 , u3 ∈ Fq , we let l1 (u1 ) := x1 (u1 )x2 (2u1 )x3 (−2u1 )x4 (u1 )x6 (u1 )x7 (2u1 )x8 (−2u1 ), l2 (u2 ) := l1 (u2 )x9 (u22 )x10 (−u22 )x11 (u22 )x14 (−u22 )x15 (2u22 ), l3 (u3 ) := l2 (u3 )x16 (4u33 )x17 (2u33 )x22 (3u33 ), and we define X12,13 := {x12 (t)x13 (−t) | t ∈ Fq } and F4 := {l3 (eu)x23 (3e4 u4 ) | u ∈ F5 },

F5 := {x5 (v/e5 ) | e ∈ F5 }.

We put V (q) := ZX12,13 F4 F5 , and for f1 = f1...,4,6,...,11,14,...,17,22,23 and f2 = f5 in a∗

the irreducible character of V (q) such that F5 , we denote by λ = λf137,...,43 ,f2 λ(x12 (t)x13 (−t)) = φ(a∗37,...,43 t),

λ|F4 = μfF14 ,

λ|F5 = μfF25 ,

and λ(xi (t)) = φ(t) for i = 37, . . . , 43. Let H := ZX{12,13} X{18,...,21} X{24,...,36} XN . Then we have a family a∗

| f1 , f2 ∈ F5 and a∗37,...,43 is a fourth power in F× F := {χf137,...,43 q }, ,f2 where

a∗

U (q)

a∗

5 F4 χf137,...,43 := IndHX5 F4 IndHX λf137,...,43 , ,f2 ,f2 V (q)

of 25(q − 1)/4 irreducible characters of degree q 16 /5. In a similar way of Proposition 5.2, we observe that a split maximal torus of G(q) acts transitively on Irr(X37 )× × · · · × Irr(X43 )× . This gives (q − 1)7 · 25(q − 1)/4 = 25(q − 1)8 /4 irreducible characters of U (q) of degree q 16 /5.  Authorship: The second author passed away on July 26th, 2018. The first and third authors are deeply grateful to him for an extraordinary collaboration experience and would like to give him credit for insight and structure of this work. They also agree on the fact that he would have concurred with the final changes to the work. Acknowledgment: We would like to thank the referees for their comments and suggestions. We would also like to thank G. Malle for his feedback on a previous version of the work. The first and third authors acknowledge financial support from the SFB-TRR 195 and the NRF Incentive Grant 109304 respectively. References [1] M. Boyarchenko and V. Drinfeld, A motivated introduction to character sheaves and the orbit method for unipotent groups in positive characteristic, arXiv:math/0609769 (2006). [2] M. Geck, G. Hiss, F. L¨ ubeck, G. Malle and G. Pfeiffer, CHEVIE – A system for computing and processing generic character tables for finite groups of Lie type, Weyl groups and Hecke algebras, Appl. Algebra Engrg. Comm. Comput. 7 (1996), 175– 210.

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[3] P. Deligne, G. Lusztig, Representations of reductive groups over finite fields, Annals of Math. 103 (1976), 103–161. [4] S. M. Goodwin, T. Le, K. Magaard and A. Paolini, Constructing characters of Sylow p-subgroups of finite Chevalley groups, J. Algebra 468 (2016), 395–439. [5] R. Gow, M. Marjoram and A. Previtali, On the irreducible characters of a Sylow 2subgroup of the finite symplectic group in characteristic 2, J. Algebra 241 (2001), no. 1, 393–409. [6] S. M. Goodwin, P. Mosch and G. R¨ohrle, On the coadjoint orbits of maximal unipotent subgroups of reductive groups, Transformation Groups (2015), 1–28. [7] M. Geck and G. Pfeiffer. Unipotent characters of the Chevalley groups D4 (q), q odd, Manuscripta Mathematica 76 (1992), no. 1, 281–304. [8] F. Himstedt, T. Le and K. Magaard, Characters of the Sylow p–subgroups of the Chevalley groups D4 (pn ), J. Algebra 332 (2011), no. 1, 414–427. [9] F. Himstedt, T. Le and K. Magaard, On the characters of the Sylow p-subgroups of untwisted Chevalley groups Yn (pa ), LMS J. Comput. Math. 19 (2016), 303–359. [10] B. Huppert, Character Theory of Finite Groups, Walter de Gruyter, Berlin, 1998. [11] I. M. Isaacs, Character theory of finite groups, Dover Books on Mathematics, New York, 1994. [12] I. M. Isaacs, Characters of groups associated with finite algebras, J. Algebra 177 (1995), 708–730. [13] I. M. Isaacs, Counting characters of upper triangular groups, J. Algebra 315 (2007), 698–719. [14] T. Le and K. Magaard, On the character degrees of Sylow p-subgroups of Chevalley groups G(pf ) of type E, Forum Math. 27 (2015), no. 1, 1–55. [15] T. Le, K. Magaard and A. Paolini, The irreducible characters of the Sylow p-subgroups of D6 (pf ) and E6 (pf ), in preparation. [16] L. Lambe and B. Srinivasan, A computation of Green functions for some classical groups. Comm. Algebra 18 (1990), no. 10, 3507–3545. [17] G. Lusztig, Character sheaves and generalizations, in: The unity of mathematics (In honor of the ninetieth birthday of I.M. Gelfand, Editors: P. Etingof, V. Retakh, I. M. Singer), Progr. Math. 244, 443–455, Birkh¨ auser Boston, Boston, MA, 2006, arXiv: math.RT/0309134. [18] M. Marjoram, Irreducible characters of a Sylow p-subgroup of the orthogonal group, Comm. Algebra 27 (1999), no. 3, 1171–1195. [19] J. Sangroniz, Character degrees of the Sylow p-subgroups of classical groups, Groups St. Andrews 2001 in Oxford, vol. 2, 487–493, London Math. Soc. Lecture Note Ser. 305, Cambridge Univ. Press, Cambridge, 2003. [20] B. Szegedy, Characters of the Borel and Sylow subgroups of classical groups, J. Algebra 267 (2003), no. 1, 130–136.

PATTERNS ON SYMMETRIC RIEMANN SURFACES ∗ and DAVID SINGERMAN† ˘ ADNAN MELEKOGLU ∗

Department of Mathematics, Faculty of Arts and Sciences, Adnan Menderes University, 09010 Aydın, Turkey Email: [email protected]

† School of Mathematics, University of Southampton, Highfield, Southampton SO17 1BJ, UK Email: [email protected]

Abstract This is a survey article mainly based on the results of [3] and [18]. We are interested in regular maps on compact symmetric Riemann surfaces. A surface is symmetric if it admits an antiholomorphic involution (symmetry). The fixed-point set of this symmetry is a collection of simple closed curves called mirrors. These mirrors pass through the vertices, edge-centres and face-centres of the map forming a sequence which we call a pattern. Klein in 1879 calculated the pattern for the Riemann surface named after him. Here we discuss the patterns for various families of Riemann surfaces, ending with the Hurwitz surfaces, these admitting 84(g − 1) automorphisms.

1

Symmetric surfaces

Every compact Riemann surface is represented by a complex algebraic curve. The surface is said to be symmetric if it admits an antiholomorphic involution (or symmetry). The symmetric Riemann surfaces correspond to real algebraic curves as first noted by Klein [12]. Example 1.1 The easiest example of a compact Riemann surface is the Riemann sphere, the unique Riemann surface of genus 0. This is symmetric, with two symmetries, the first complex conjugation given by z → z¯ and the second the antipodal map given by z → −1/¯ z . These are topologically distinct as the first has one fixedcurve namely the equator while the second is fixed-point free. Example 1.2 (Klein’s Riemann surface of genus 3) This is one of the most famous examples of compact Riemann surfaces, and perhaps one of the most interesting for group theorists, as its conformal automorphism group is PSL(2, 7). It is the Riemann surface of the algebraic curve known as the Klein quartic which has the homogeneous equation x3 y + y 3 z + z 3 x = 0. It is also the compactification of H/Γ(7), where H is the upper half-plane and Γ(7) is the principal congruence subgroup mod 7 of the modular group.

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Figure 1. Klein quartic Klein’s surface is visibly symmetric as reflection in the imaginary axis is a symmetry. See Figure 1. (Klein gave the following edge pairings: 1–6, 2–11, 3–8, 4–13, 5–10, 7–12, 9–14.) Also it is symmetric because it is a real curve! Definition 1.3 A fixed-curve of a symmetry is called a mirror.

2

Maps and Regular maps

A map on a surface X is an embedding of a graph G into X with the property that X \ G is a union of two-cells. Thus, a map has vertices, edges and faces. The vertices and edges come from the graph and the faces are the components of X \ G. A directed edge is called a dart and the map is called regular if its automorphism group is transitive on its darts. (By an automorphism here, we mean an orientation-preserving automorphism.) Some regular maps admit reflections which are orientation-reversing transformations that preserve an edge, and interchange the two darts of that edge without interchanging the two faces adjacent to that edge. A regular map that admits such a reflection is called reflexible. A map is said to have type {m, n} if m is the face valency (size) and n is the vertex valency. 2.1

Triangle groups

Let T be a triangle with angles π/2, π/m and π/n. Then T lies on the sphere, the Euclidean plane or the hyperbolic plane depending on whether (1/m) + (1/n) greater than, equal to or less than 1/2. We call such a triangle a (2, m, n)-triangle. Let P , Q and R denote the reflections in the sides of T , as indicated in Figure 2.

Meleko˘ glu, Singerman: Patterns on Symmetric Riemann Surfaces ......... ... ........... .. .... ............ ....... ............ ....... ... ....... ... ....... ....... .... ....... ....... ... ....... ....... ..... ....... ....... .... ........ ....... ... ........ .. . . ... .............. .............. ... .. ...................................................................................................................................................................

π/n

P

445

R

π/2

π/m

Q

Figure 2. A (2, m, n)-triangle These reflections satisfy the relations P 2 = Q2 = R2 = (P Q)2 = (QR)m = (RP )n = 1.

(1)

The triangle group Γ[2, m, n] is generated by A = P Q, B = QR and C = RP obeying the relations A2 = B m = C n = ABC = 1.

3

(2)

Maps and Riemann surfaces

It is known that underlying every map on an orientable surface there is associated a natural complex structure that makes this surface into a Riemann surface in such a way that every automorphism of the map becomes an automorphism of the Riemann surface. (See [7, 9, 20], though in [7] maps are called dessin d’enfants.) In particular, every reflection of the map becomes a symmetry of the Riemann surface. (For many years this result was folklore; for example Klein associated a map with the Klein quartic and in [5, Chapter 8] Coxeter and Moser associate maps with hyperelliptic surfaces.) What are the Riemann surfaces that are associated to maps? Grothendieck noted that Bely˘ı’s theorem implies that the algebraic curve associated to a Remann surface underlying a map is defined over the field of algebraic numbers – see [7, 9], for example. (The second author first heard about Bely˘ı’s theorem at the St Andrews conference in 1989 when John Thompson introduced it in his talk on Galois groups. Unfortunately, many of us failed to realise its full significance then!)

4

Automorphisms of Riemann surfaces

The set of all conformal automorphisms of a Riemann surface X form a group Aut+ (X), which is a subgroup of index 1 or 2 in the group of all automorphisms, including those that reverse orientation, denoted by Aut± (X). If the genus g of X is greater than 1, then by a theorem of Hurwitz, Aut+ (X) is a finite group whose order is bounded above by 84(g − 1). If X has this number of automorphisms, then it is called a Hurwitz surface and Aut+ (X) is called a Hurwitz group. The Klein surface of genus 3 is the Hurwitz surface of smallest genus. Hurwitz groups are finite homomorphic images of the [2, 3, 7] triangle group. Indeed if θ : Γ[2, 3, 7] → G is a group epimorphism, then its kernel K is the fundamental group of a Hurwitz surface X with Aut+ (X) isomorphic to G. In 1961, Macbeath observed that we can

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Meleko˘ glu, Singerman: Patterns on Symmetric Riemann Surfaces

construct infinitely many Hurwitz groups by observing that [K, K]K n is a normal (surface) subgroup of Γ for all positive integers n. This is sometimes called the Macbeath trick but the idea goes back to Burnside. These give surfaces of very high genus but Macbeath had another useful method of getting Hurwitz groups of smaller genus, by finding which groups PSL(2, q) are finite images of the [2, 3, 7] triangle group. His result [16] is as follows. PSL(2, q) is a Hurwitz group if and only if (i) q = 7, or (ii) q = p for some prime p ≡ ±1 mod 7, or (iii) q = p3 for some prime p ≡ ±1 mod 7. In cases (i) and (iii) there is exactly one Hurwitz surface but in case (ii) there are three different Hurwitz surfaces.

5

Patterns

Let M be a regular map on a Riemann surface X and let M be a mirror on Then M passes through some geometric points of M. By geometric points mean the vertices, the edge-centres and the face-centres of M. Following [4], denote these vertices, edge-centres and face-centres by 0, 1 and 2, respectively. the geometric points on M form a periodic sequence of the form

X. we we So

a1 a2 . . . ak−1 ak a1 a2 . . . ak−1 ak . . . a1 a2 . . . ak−1 ak        

(3)

1

2

N

which we call the pattern of M , where ai ∈ {0, 1, 2} and 1 ≤ i ≤ k. We call a1 a2 . . . ak−1 ak of (3) a link of the pattern and call N the link index. We abbreviate the pattern (3) to (a1 a2 . . . ak−1 ak )N . As an example consider the dodecahedral map on the Riemann sphere in Figure 3. The reflection z → z¯ fixes the equator which passes through vertices, edgecentres and face-centres giving the pattern 010212010212, which we abbreviate to (010212)2 ; see Figure 3. In Coxeter’s “Regular Polytopes” [4], patterns for all the regular maps on the sphere are given. However well before Coxeter, Felix Klein had discussed this very idea in his famous paper [11] where he introduced the Klein quartic. This corresponds to his Riemann surface of genus 3 with automorphism group PSL(2, 7) of order 168. Associated with this surface is the well-known map of type {3, 7}. Klein’s surface is symmetric and in Section 13 of his paper Klein points out that on his surface a mirror has pattern (010212)3 . So for the Klein quartic the link is 010212 and the link index is 3. (However in Klein’s paper (see [11, p465] or [13, p322]) he uses a, b, c for 2, 0, 1.) Our aim is to examine the links and link indices for all regular maps.

6

Universal maps

Our technique is to lift every map on a compact surface to one of the universal maps on the sphere, plane or hyperbolic plane. (These are the simply-connected Riemann surfaces.) We then find the patterns for these. On the sphere these are just the patterns for the platonic solids such as the dodecahedron and these were known to Coxeter and listed also in [18]. There is a universal map of type

Meleko˘ glu, Singerman: Patterns on Symmetric Riemann Surfaces

447

0 •.... ... 1 •... ... 0 •.. ... ... ... ... 2 •... ... ... 1 •.... ... ... 2 •.. ... •. 0

..................................................................... ...... .... .. ........... ...... ... .. ...... ...... ... .. ...... ...... . . . . ... .. ...... . . .. ...... ... ...... . . . . . ... ... ...... .. ... . .... ...... ...... .. .... ...... .... ......................... . . . . . . .... . . ... . . .... ...... ........... ....................... . . ... .... . . . . . . .... .................. ............. ... . . . . . .... ... . . ... ... ... ..... ... ..... ... ... ..... . . ... . . . . . ... . ...... ... ..... . . . . .... .. ........................... ............................................. ... . . . ..... ... . ........ .... ... . ... ..... .. ........ ... . . . . ... .. . .. ..... ... ... .... ... .. . .. ... ..................... .................. ... ...... ... ........... ......................... .... ... ...... .... ... .......................... ...... .... ... ...... . .. . . . . ....... . .. ....... ... .. ...... ... ..... .. ...... ... ...... .. ...... .. ..... .. ...... ...... ... .. ...... .. ...... . ...... .... . . . . ... ...... ... ............................................................

Figure 3. Dodecahedron {m, n}, and this lies on the hyperbolic plane (usually represented by the upper half-plane H), the Euclidean plane C or the sphere Σ. Except for the last case, the patterns are infinite. Everything depends on the parities of m, n. For example if m, n are odd the pattern is (010212)∞ while if m, n are both even then there are three patterns (01)∞ , (12)∞ , (02)∞ . You can see this easily by looking at the chessboard tessellation of type {4, 4}. Remark 6.1 As shown in [18], the pattern of a mirror is obtained from one of the following links: 12, 02, 01, 0102, 0212, 010212. Patterns for the universal maps are given in Table 1. For finite universal maps, i.e., spherical maps we get the same patterns but the link indices are now positive integers. Table 1. Patterns of universal maps Case Reflections Pattern

7

m and n odd

P , Q, R

(010212)∞

m odd n even m odd n even

P Q, R

(01)∞ (0212)∞

m even n odd m even n odd

Q P, R

(12)∞ (0102)∞

m and n even m and n even m and n even

P Q R

(01)∞ (12)∞ (02)∞

Mirror automorphisms

Definition 7.1 Let X be a Riemann surface with underlying map M and let M be a mirror of a reflection of M. A mirror automorphism of M is an orientation-

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preserving automorphism of M that cyclically permutes the links of the pattern of M . Mirror automorphisms associated to particular patterns are given in Table 2. This result comes from [18]. Table 2. Patterns and mirror automorphisms Link Mirror Automorphism 01 02 12 0102 0212 010212

C n/2 A B m/2 C n/2 B m/2 A C (n+1)/2 BC (n+1)/2 B m/2 C n/2 B (m+1)/2 CB (m+1)/2 B (m+1)/2 CB (m+1)/2 C (n+1)/2 BC (n+1)/2

The following result is useful in determining the link index for a general regular map. Lemma 7.2 Let T be a symmetry of a Riemann surface with mirror M . Associated to M we have a pattern π with link index K and a mirror automorphism SM whose order is equal to K. It is known that a regular map of genus one is either of type {4, 4}, {3, 6} or {6, 3}. In [5] these are classified as the maps {4, 4}b,c , {3, 6}b,c and {6, 3}b,c . These exist for all pairs of integers (b, c) = (0, 0). So there are infinitely many regular maps of genus g = 1. The calculations of their patterns are fairly straightforward; see [18].

8

Surfaces of genus g > 1

For each g > 1, there are only finitely many regular maps of genus g. We only consider some particularly interesting examples such as Accola-Maclachlan, Fermat and Hurwitz maps. 8.1

Accola-Maclachlan maps

Let μ(g) be the maximum number of conformal automorphisms of all Riemann surfaces of genus g > 1. It was shown independently by Accola [1] and Maclachlan [17] that μ(g) ≥ 8(g + 1), and for every g ≥ 2 there is a Riemann surface of genus g with 8(g + 1) conformal automorphisms. These surfaces are known as AccolaMaclachlan surfaces. The Accola-Maclachlan surface of genus g is the two-sheeted covering of the sphere branched over the vertices of a regular (2g + 2)-gon. It underlies a regular map M of type {2g + 2, 4} called the Accola-Maclachlan map.

Meleko˘ glu, Singerman: Patterns on Symmetric Riemann Surfaces

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Theorem 8.1 Let M be the Accola-Maclachlan map of g > 1. Then the patterns of M are either (01)2 , (02)2 and (12)2 if g is odd, or (01)2 , (02)2 and (12)4 if g is even. 8.2

Fermat maps

The Fermat curve Fn is the Riemann surface of the projective algebraic curve {(x, y, z) | xn + y n + z n = 0}, in P2 (C). Consider the triangle group Γ[n, n, n], which has the presentation x, y, z | xn = y n = z n = xyz = 1 . If we abelianize this group, we get Zn ⊕ Zn so that the commutator subgroup Kn of Γ[n, n, n] is a characteristic subgroup of index n2 . So there is a homomorphism θ : Γ[n, n, n] → Zn ⊕ Zn whose kernel is Kn with H/Kn ∼ = Fn ; see [10]. Now Γ[n, n, n]  Γ[2, 3, 2n] ([19]) with Γ[2, 3, 2n]/Γ[n, n, n] ∼ = S3 (noting that Γ[2, 3, 2] ∼ = S3 ). We can extend the homomorphism θ to θ∗ : Γ[2, 3, 2n] → (Zn ⊕ Zn )  S3 so the kernel of θ∗ is also Kn . Thus, Kn is normal in Γ[2, 3, 2n] with index 6n2 and hence there is a regular map of type {3, 2n} on Fn called the Fermat map of degree n. We then find from the Riemann-Hurwitz formula that Fn has genus (n − 1)(n − 2)/2. As Kn is characteristic in Γ[n, n, n] and as Γ[n, n, n] is normal in Γ[2, 3, 2n], Kn is normal in Γ[2, 3, 2n] and Aut+ (Fn ) ∼ = (Zn ⊕ Zn )  S3 . = Γ[2, 3, 2n]/Kn ∼ The following theorem was proved in [18]. Theorem 8.2 Let Mn be the Fermat map of degree n. Then: (i) If n if odd, then the patterns of Mn have one of the forms (01)3 and (0212)n ; (ii) If n if even, then the patterns of Mn have one of the forms (01)4 and (0212)2 . 8.3

Hurwitz maps

A Hurwitz map M is a map of type {3, 7} that is associated with a Hurwitz surface X. It is known that Aut+ (M) has a presentation of the form A, B, C | A2 = B 3 = C 7 = ABC = · · · = 1 . Let S be a mirror automorphism of M, which by Table 2 we can choose S = B 2 CB 2 C 4 BC 4 . If S has order 1, then by using MAGMA [2] we find that A, B, C | A2 = B 3 = C 7 = ABC = B 2 CB 2 C 4 BC 4 = 1 gives the trivial group! This shows that a Hurwitz map cannot have a link index equal to 1. Similarly, A, B, C | A2 = B 3 = C 7 = ABC = S 2 = 1

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is a presentation for a group of order 504. So the genus must be 7 and X is the Fricke-Macbeath surface; see [6, 15]. In the same way, we see that A, B, C | A2 = B 3 = C 7 = ABC = S 3 = 1 gives a group of order 168. So X is Klein’s surface of genus 3. This gives the following interesting result. Theorem 8.3

(i) A Hurwitz map cannot have link index 1.

(ii) A Hurwitz map has link index 2 if and only if the underlying Riemann surface is the Fricke-Macbeath surface of genus 7. (iii) A Hurwitz map has link index 3 if and only if the underlying Riemann surface is Klein’s surface of genus 3. As a consequence, the minimum length of a mirror on a Hurwitz surface occurs for the Fricke-Macbeath surface. Using Theorem 8.3 together with Macbeath’s trick (§6) the following result can be obtained, which appears in [3]. Theorem 8.4 For every positive integer n, there exist Hurwitz maps with link indices 2n and 3n. In particular, the link index of a Hurwitz map can be any even positive integer. Thus, every even integer is the link index of some Hurwitz map. In [3], it is shown that every odd integer 1 < n < 361 is the link index of some Hurwitz map. However, in some cases it is not easy to find. For example, link index 17 was first found for genus 8438382031813; see Table 3. We also list the smaller Hurwitz maps corresponding to the Macbeath surfaces with PSL(2, q) automorphism groups and Hurwitz maps of genus up to 1000, together with their link indices. (See Tables 4 and 5). We note that when there are three distinct surfaces of the same genus with automorphism group PSL(2, q), (q ≡ ±1 mod 7) we can have different link indices for the same map. The information in Tables 3, 4 and 5 are taken from larger tables which appear in [3]. Acknowledgement We would like to thank the referee for carefully reading our work and for making some helpful comments.

Meleko˘ glu, Singerman: Patterns on Symmetric Riemann Surfaces

Table 3. Examples of Hurwitz maps with given odd link index < 58 Link index

Genus

Aut+ (M)

3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57

3 411 14 2131 2091 118 146 8438382031813 8589 411 15986 70!/168 + 1 29!/168 + 1 29!/168 + 1 63!/168 + 1 15!/168 + 1 2131 66009 43!/168 + 1 3404 86!/168 + 1 35!/168 + 1 132071 5433 57!/168 + 1 57!/168 + 1 42!/168 + 1 8589

PSL(2, 7) PSL(2, 41) PSL(2, 13) PSL(2, 71) J1 PSL(2, 27) PSL(2, 29) PSL(2, 112337) PSL(2, 113) PSL(2, 41) PSL(2, 139) A70 A29 A29 A63 A15 PSL(2, 71) PSL(2, 223) A43 PSL(2, 83) A86 A35 PSL(2, 281) PSL(2, 97) A57 A57 A42 PSL(2, 113)

451

452

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Table 4. Link indices of Hurwitz maps with conformal automorphism group PSL(2, q) for small q Aut+ (M)

Genus

Reflexible?

Link indices

PSL(2, 7) PSL(2, 8) PSL(2, 13) PSL(2, 27) PSL(2, 29) PSL(2, 41) PSL(2, 43) PSL(2, 71) PSL(2, 83) PSL(2, 97) PSL(2, 113) PSL(2, 125)

3 7 14 118 146 411 474 2131 3404 5433 8589 11626

Yes (1 map) Yes (1 map) Yes (3 maps) Yes (1 map) Yes (3 maps) Yes (3 maps) Yes (3 maps) Yes (3 maps) Yes (3 maps) Yes (3 maps) Yes (3 maps) Yes (1 map)

3 2 6, 7, 7 13 14, 15, 15 5, 7, 21 21, 22, 22 9, 18, 35 14, 41, 42 8, 49, 49 19, 28, 57 62

Meleko˘ glu, Singerman: Patterns on Symmetric Riemann Surfaces

Table 5. Link indices of Hurwitz maps of genus less than 1000 Map

Genus

|Aut+ (M)|

Reflexible?

Link index

H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15 H16 H17 H18 H19 H20 H21 H22 H23 H24 H25

3 7 14 14 14 17 17 118 129 129 129 146 146 146 385 411 411 411 474 474 474 687 769 769 769

168 504 1092 1092 1092 1344 1344 9828 10752 10752 10752 12180 12180 12180 32256 34440 34440 34440 39732 39732 39732 57624 64512 64512 64512

Yes Yes Yes Yes Yes No No Yes Yes No No Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes

3 2 6 7 7 6 6 13 6 12 12 14 15 15 4 5 7 21 21 22 22 21 4 4 4

453

454

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References [1] R. D. M. Accola, On the number of automorphisms of a closed Riemann surface, Trans. Amer. Math. Soc. 131 (1968), 398–408. [2] W. Bosma, J. Cannon and C. Playoust: The Magma Algebra System I: The User Language, J. Symbolic Comput. 24 (1997), 235–265. [3] M. Conder and A. Meleko˘glu, Link indices of Hurwitz maps, J. Algebra 490 (2017), 568–580. [4] H. S. M. Coxeter, Regular Polytopes (Dover Publications, New York 1973). [5] H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups (Springer-Verlag, Berlin, Heidelberg 1980). ¨ [6] R. Fricke, Uber eine einfache Gruppe von 504 Operationen, Math. Ann. 52 (1899), 321–339. [7] A. Grothendieck, Esquisse d’un Programme. in Around Grothendieck’s Esquisse d’un Programme (L. Schneps, P. Lochak eds.) London Math. Soc. Lecture Note Ser. 242, (CUP, Cambridge, 1997), 5–48. ¨ [8] A. Hurwitz, Uber algebraische Gebilde mit eindeutigen Transformationen in sich, Math. Ann. 41 (1893), 403–442. [9] G. A. Jones and D. Singerman, Theory of maps on orientable surfaces, Proc. London Math. Soc. (3) 37 (1978), 273–307. [10] G. A. Jones and D. Singerman, Bely˘ı functions hypermaps and Galois groups, Bull. Lond. Math. Soc. 28(6) (1996), 561–590. ¨ [11] F. Klein, Uber die Transformation siebenter Ordnung der elliptischen Funktionen, Math. Ann. 14 (1879), 428–471. [12] F. Klein, On Riemann’s theory of algebraic functions and their integrals (Dover Publications, New York 1963). [13] S. Levy, On the order-seven transformation of elliptic functions, in The eightfold way: The beauty of Klein’s quartic curve. (S. Levy, eds.) Mathematical Sciences Research Institute Publications 35, (CUP, Cambridge 1999), 287–331. [14] A. M. Macbeath, On a theorem of Hurwitz, Proc. Glasgow Math. Assoc. 5 (1961), 90–96. [15] A. M. Macbeath, On a curve of genus 7, Proc. London Math. Soc. 15 (1965), 527–542. [16] A. M. Macbeath, Generators of the linear fractional groups, Number Theory, Proc. Sympos. Pure Math., vol. XII, (Amer. Math. Soc., Providence, R.I., 1969), 14-32. [17] C. Maclachlan, A bound for the number of automorphisms of a compact Riemann surface, J. London Math. Soc. 44 (1969), 265–272. [18] A. Meleko˘glu and D. Singerman, The structure of mirrors on regular maps on Platonic surfaces, Geom. Dedicata 181 (2016), 239–256. [19] D. Singerman, Finitely maximal Fuchsian groups. J. London Math. Soc. 6 (1972), 29–38. [20] D. Singerman, Automorphisms of maps, permutation groups and Riemann surfaces, Bull. London Math. Soc. 8 (1976), 65–68.

SUBGROUPS OF TWISTED WREATH PRODUCTS ´ ´ PETER P. PALFY Alfr´ed R´enyi Institute of Mathematics, Hungarian Academy of Sciences Re´ altanoda utca 13–15, 1053 Budapest, Hungary Email: [email protected]

Abstract By determining subdirect products invariant under the action of a regular permutation group of the components we provide a natural motivation for the definition of twisted wreath products. Then—based on papers of R. Baddeley, A. Lucchini, F. B¨ orner, and M. Aschbacher—we explain how twisted wreath products play a fundamental role in the problem of representing finite lattices as intervals in subgroup lattices of finite groups.

1

Introduction

Our first goal is to provide a natural motivation for the definition of twisted wreath products. Originally, the twisted wreath product was introduced by Bernhard H. Neumann [17] in 1963. At first glance his definition looks quite complicated. Michio Suzuki [22, Chapter 2, §10] presented a more elegant treatment of this construction. In Section 2 we will determine all those subdirect products in a direct product of isomorphic non-abelian simple groups that are invariant under a regular permutation group of the components. This naturally leads to the definition of the twisted wreath product. Twisted wreath products occur in the O’Nan–Scott–Aschbacher Theorem on the classification of primitive finite permutation groups. They were erroneously omitted from the first version [20] of the theorem, and were only added later to the list in the paper of Michael Aschbacher and Leonard Scott [5], and independently by L´aszl´ o Kov´ acs [14]. (See also [16].) Although in the original paper of B. H. Neumann [17], as well as in several later developments, twisted wreath products were used for the construction of infinite groups with certain peculiar properties, in the present paper we will restrict our attention to twisted wreath products of finite groups. Our second goal is to explain the role twisted wreath products play for the problem of representing finite lattices as intervals in subgroup lattices of finite groups. This was explicitly or implicitly observed in the papers of Robert Baddeley and Andrea Lucchini [7], Baddeley [6], Ferdinand B¨ orner [8], and Michael Aschbacher [2]. Based on their results we present in Section 3 a simplified proof showing that this representation problem can be reduced either to the case of almost simple groups or to the case of twisted wreath products. In Section 4 we give proper credits to the original papers and make further comments on the related literature.

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We tried to make the paper as much self-contained as possible. However, at several places the proof makes use of Schreier’s Hypothesis claiming that the outer automorphism group Out(T ) = Aut(T )/Inn(T ) of every finite non-abelian simple group T is solvable. This is a well-known consequence of the classification of finite simple groups. As for many questions in finite group theory it would be desirable to reduce the problem to the case of almost simple groups (groups G with a simple normal subgroup T with CG (T ) = 1). However, it seems inevitable to consider also certain twisted wreath products in the context of representing finite lattices as intervals in subgroup lattices of finite groups. The basic group theoretic notions do not need explanation for the readership of this proceedings. As lattice theory is concerned, let us recall that a lattice L is a partially ordered set where any two elements x, y have a greatest lower bound (called their meet, denoted by x ∧ y) and a least upper bound (their join, denoted by x∨y). Finite lattices have a smallest and a largest element; these will be denoted by 0L and 1L . By a filter F in a finite lattice L we mean a non-empty subset of the form {x ∈ L | x ≥ a} for some a ∈ L. We obtain the dual of a lattice when we reverse the ordering, so the meet in the dual lattice is the same as the join in the original lattice, and similarly, the new join is the old meet. A lattice L is called modular if ∀x, y, z ∈ L : x ≤ z ⇒ (x ∨ y) ∧ z = x ∨ (y ∧ z). The subgroup lattice of an abelian group is always modular. We will call a lattice L consisting of more than two elements strongly non-modular if for every y ∈ L, y = 0L , 1L , there exists a pair of elements x < z ∈ L such that y ∨ x = y ∨ z and y ∧ x = y ∧ z.

y u

J

2

u y∨x=y∨z

@ @ @u z

J

ux JJu y∧x=y∧z

Invariant subdirect products

Let T (the “target”) be a finite non-abelian simple group and let D (the “domain”) be an arbitrary finite group. Consider the group F of all functions D → T with pointwise multiplication. Then F ∼ = T × · · · × T = T |D| . Let D act on F by translation, that is for f ∈ F , d ∈ D let f d ∈ F be the function f d (x) = f (xd−1 ) (x ∈ D). Clearly, this defines an action of the group D on F , as we have f d1 d2 (x) = −1 −1 d1 d1 d2 f (x(d1 d2 )−1 ) = f ((xd−1 2 )d1 ) = f (xd2 ) = (f ) (x). The semidirect product F  D is the regular wreath product T  D. Recall that a subgroup H ≤ G1 × · · · × Gn of a direct product is said to be a subdirect product, if the projection of H to each factor Gi (i = 1, . . . , n) is

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surjective. If the factors are pairwise isomorphic non-abelian simple groups, then the structure of any subdirect product is described by the following lemma (see, e.g., [9, Exercise 4.3]). Lemma 2.1 Let T be a non-abelian simple group and let H ≤ T n be a subdirect product. Then H ∼ = T m for some 1 ≤ m ≤ n. Moreover, there exist a map ν : {1, . . . , n} {1, . . . , m} and automorphisms ϕi ∈ Aut(T ) such that f : {1, . . . , n} → T belongs to H iff f (i) = ϕi (tν(i) ) with t1 , . . . , tm ∈ T . Now we are going to determine which subdirect products in F ∼ = T × ··· × T are invariant under the action of D. Let H ≤ F be a subdirect product. By the lemma we have H ∼ = T m for some 1 ≤ m ≤ |D|, and H = {f : D → T | f (x) = ϕx (tν(x) ), t1 , . . . , tm ∈ T }, with appropriate ν : D {1, . . . , m} and ϕx ∈ Aut(T ) (x ∈ D). For f ∈ H, −1 b ∈ D, the invariance of H means that f b also belongs to H, hence with some u1 , . . . , um ∈ T we have −1 f b (x) = ϕx (uν(x) ). By the definition of the action of D we obtain −1

f b (x) = f (xb) = ϕxb (tν(xb) ). Clearly, D preserves the partition given by the kernel of the map ν, hence it is a partition into the cosets of some subgroup D0 ≤ D: D = D0 x1 ∪ D0 x2 ∪ · · · ∪ D0 xm with ν(d) = i iff d ∈ D0 xi . Without loss of generality we may assume that x1 = 1 and that for every i = 1, . . . , m we have ϕxi = id. If b ∈ D0 xi and a ∈ D0 , then ab ∈ −1 D0 xi as well. Then f (ab) = f b (a) = ϕa (uν(a) ) = ϕa (u1 ). For a = 1 this yields u1 = f (b), hence ∀a ∈ D0 , ∀b ∈ D : f (ab) = ϕa (f (b)). If b = 1 we get ∀a ∈ D0 : f (a) = ϕa (f (1)). If a, b ∈ D0 , then ϕab (f (1)) = f (ab) = ϕa (f (b)) = ϕa (ϕb (f (1))). Since H is a subdirect product, f (1) can be any element of T , so ϕ : D0 → Aut(T ) is a homomorphism. Furthermore, ϕaxi (ti ) = f (axi ) = ϕa (f (xi )) = ϕa (ti ), so ∀a ∈ D0 , ∀i ∈ {1, . . . , m} : ϕaxi = ϕa . Conversely, it is easy to verify that if D0 ≤ D, D = D0 x1 ∪ · · · ∪ D0 xm , and ϕ : D0 → Aut(T ) is a homomorphism, then by defining Sdp(D0 , ϕ) = {f : D → T | f (axi ) = ϕa (ti ), a ∈ D0 , ti ∈ T (i = 1, . . . , m)} ∼ T |D| . Indeed, if b ∈ D, then we obtain a D-invariant subdirect product in F = multiplication by b from the right permutes the right cosets of D0 and so we have xi b = ai xj with ai ∈ D0 , 1 ≤ j = j(i, b) ≤ m. If f ∈ Sdp(D0 , ϕ) let us denote f (xi b) −1 by ui . With this notation we have f b (axi ) = f (axi b) = f (aai xj ) = ϕaai (tj ) = −1 ϕa (ϕai (tj )) = ϕa (f (ai xj )) = ϕa (f (xi b)) = ϕa (ui ), so f b ∈ Sdp(D0 , ϕ), as we wanted. Thus we have proved the following proposition.

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Proposition 2.2 If T is a non-abelian simple group, D is any finite group, and F = {f : D → T }, then the D-invariant subdirect products in F ∼ = T |D| are precisely the subgroups of the form Sdp(D0 , ϕ) for subgroups D0 ≤ D and homomorphisms ϕ : D0 → Aut(T ). We can define Sdp(D0 , ϕ) even without assuming the simplicity of T , the construction makes sense for any T . Then the semidirect product Sdp(D0 , ϕ)  D is the twisted wreath product Twr(T, D, D0 , ϕ) of T and D with respect to the subgroup D0 ≤ D and the homomorphism ϕ : D0 → Aut(T ). If we want to compare invariant subdirect products, the following is obvious. Lemma 2.3 A subdirect product Sdp(D1 , ϕ1 ) is contained in another subdirect product Sdp(D2 , ϕ2 ) iff D1 contains D2 and ϕ2 is the restriction of ϕ1 to D2 . Now we return to analyzing the subgroup structure of F when T is a non-abelian simple group. Lemma 2.4 If T is a non-abelian simple group, and D0 ≤ D, ϕ : D0 → Aut(T ) satisfy ϕ(D0 ) ≥ Inn(T ), then every non-trivial D-invariant subgroup of Sdp(D0 , ϕ) is a subdirect product. Proof Let H ≤ Sdp(D0 , ϕ) be any D-invariant subgroup. Consider the image U = {f (1) | f ∈ H} of H under the projection to the first coordinate. Let u ∈ U (so u = f (1) for some f ∈ H), and take an a ∈ D0 . Then ϕa (u) = ϕa (f (1)) = −1 f (a) = f a (1) ∈ U , hence U is a ϕ(D0 )-invariant subgroup of the simple group T . By assumption, ϕ(D0 ) ≥ Inn(T ), so U is a normal subgroup of T , hence by the simplicity of T , either U = 1 or U = T . Since D acts transitively on the components of the direct product, either all projections of H have trivial image, and so H = 1, or all projections map onto T , and so H is a subdirect product.  Thus the D-invariant subgroups of Sdp(D0 , ϕ) apart 9from the trivial subgroup are all of the form Sdp(D1 , ϕ1 ) with D0 ≤ D1 and ϕ1 9D0 = ϕ. Combining the previous two lemmas, we obtain the following. Proposition 2.5 Let T be a non-abelian simple group, and assume that D0 ≤ D, ϕ : D0 → Aut(T ) satisfy ϕ(D0 ) ≥ Inn(T ). Then the lattice of D-invariant subgroups of Sdp(D0 , ϕ) is isomorphic to the dual of the lattice of all extensions of ϕ to subgroups containing D0 , together with an additional top element (what corresponds to the trivial subgroup via the dual isomorphism). Remark 2.6 Note that in the setting of (2.5) if ϕ1 : D1 → Aut(T ) extends ϕ : D0 → Aut(T ), then ϕ1 is uniquely determined by its kernel. Indeed, if K = ker(ϕ1 ), then D1 /K ∼ = ϕ1 (D1 ) ≤ Aut(T ) is an almost simple group, and the image of D0 contains Inn(T ) ∼ = T , hence the action of D1 /K on ϕ−1 (Inn(T ))/K ∼ = T determines the homomorphism ϕ1 : D1 → Aut(T ).

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Example 2.7 Let A5 and S5 denote the alternating and the symmetric group of degree 5, and let T = A5 , D = S5 × A5 , D0 = diag(A5 ) = {(x, x) | x ∈ A5 } < D, and ϕ : D0 ∼ = A5 → Aut(T ) ∼ = S5 an embedding. It is easy to see that the subgroups of D containing D0 are D0 = diag(A5 ), A5 × A5 , and D = S5 × A5 . Now ϕ has two extensions to A5 × A5 , corresponding to the first and the second projection. Likewise, there are two extensions to S5 × A5 . Together with the additional top element this gives a hexagon lattice (here a, b ∈ A5 , s ∈ S5 ): u

TOP @ @ @u (s, b) → b (s, b) → s u (a, b) → a u

@

u (a, b) → b @

(a, a) → a@u Hence by (2.5) the lattice of D-invariant subgroups of Sdp(D0 .ϕ) is the hexagon lattice. If M is a minimal normal subgroup of a finite group, then M is characteristically simple, so it is either an elementary abelian p-group for some prime number p, or it is isomorphic to a direct power of a non-abelian simple group T . We consider the latter case, when M = T1 × · · · × Tk (k ≥ 1) and each Ti ∼ = T . Now let a group A act on T1 × · · · × Tk in such a way that A permutes the direct factors transitively. Let A1 = {a ∈ A | T1a = T1 } and denote by α the homomorphism A1 → Aut(T1 ) determined by the action of A1 . Furthermore, choose a set of coset representatives x1 = 1, x2 , . . . , xk of A1 in A and fix the isomorphism t → txi (t ∈ T1 ) between T1 = T and Ti (i = 1, . . . , k). Then Sdp(A1 , α) ∼ = T k with the isomorphism given by the projection onto the group of functions {f : {x1 , . . . , xk } → T } and this isomorphism is compatible with the action of A. In the rest of the paper we will freely use this identification of Sdp(A1 , α) and T k . (Cf. [15].) The following lemmas will be needed in the proof of the main result (3.2). Lemma 2.8 Every A-invariant subgroup of T k is one of the following types: 1. a subdirect product in T k ; 2. a box, that is, U k for some A1 -invariant subgroup 1 < U < T ; 3. a skew subgroup, i.e., a nontrivial subgroup properly contained in a box; 4. the trivial subgroup. Proof Let H ≤ T k . Let the projection to the first component map H onto U ≤ T . Since H is A-invariant, each projection maps H onto U in the corresponding component (isomorphic to T via the fixed isomorphism). So H ≤ U k . If U = T , then H is a subdirect product. If 1 < U < T and H = U k , then H is a box. If 1 < U < T and H < U k , then H is a skew subgroup. (Note that the box U k is also A-invariant in this case.) If U = 1, then H = 1 as well. 

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Corollary 2.9 A maximal A-invariant subgroup is either a subdirect product or a box, unless the only proper A-invariant subgroup is the trivial one. The following is obvious. Lemma 2.10 The box subgroups in SubA (T k ) form a sublattice isomorphic to SubA1 (T ). The next result is a well-known consequence of Schreier’s Hypothesis, see [6, Proposition 2.4]. Lemma 2.11 If α(A1 ) ≥ Inn(T ), then there exists a proper non-trivial A1 -invariant subgroup of T . Lemma 2.12 A box cannot be contained in a proper subdirect product. Proof For any proper subdirect product H there exist a pair of indices 1 ≤ i < j ≤ k and ϕ ∈ Aut(T ) such that for any (x1 , . . . , xk ) ∈ H we have xj = ϕ(xi ). Hence if xi = 1, then xj = 1 as well. In the contrary, any box subgroup contains elements with xi = 1, xj = 1.  Lemma 2.13 Let H1 > H2 be A-invariant subdirect products and let U be an A1 -invariant subgroup. Then either H1 ∩ U k > H2 ∩ U k or H2 ∩ U k = 1. 9 Proof Let H2 = Sdp(B2 , β) and H1 = Sdp(B1 , β 9B1 ) with B1 < B2 (see (2.3)). Suppose that H2 ∩ U k = 1. By the definition of H2 = Sdp(B2 , β) this means that there exists t1 , . . . , tm ∈ T (m = |A : B2 |) not all equal to 1, such that βb (ti ) ∈ U for every b ∈ B2 and each i = 1, . . . , m. In particular, we can choose 1 = u ∈ U such that βb (u) ∈ U for every b ∈ B2 . Then the function f : A → T given 9 by f (a) = βa (u), if a ∈ B1 , and f (a) = 1, otherwise, belongs to H1 = Sdp(B1 , β 9B1 ),  but not to H2 , hence H1 ∩ U k > H2 ∩ U k .

3

Intervals in subgroup lattices of finite groups

Let Sub(G) denote the subgroup lattice of the group G, and for a subgroup H < G we denote by Int(H, G) = {X | H ≤ X ≤ G} the lattice of intermediate subgroups (in other words: overgroups of H), and call it the interval between H and G in the subgroup lattice. If a group A acts by automorphisms on G, then the A-invariant subgroups form a sublattice in Sub(G); it will be denoted by SubA (G). Likewise, we will use the notation IntA (H, G), whenever H is an A-invariant subgroup of G. Intervals of subgroup lattices occur in various contexts. If F ⊂ E is a finite separable field extension and E ∗ is the splitting field containing E, then the lattice of intermediate fields {X | F ⊆ X ⊆ E} is dually isomorphic to the interval Int(Gal(E ∗ |E), Gal(E ∗ |F )) in the Galois group of E ∗ . In the theory of operator algebras it is an open problem whether every finite lattice is isomorphic to the lattice of intermediate subfactors of a von Neumann algebra. Yasuo Watatani [24] proved that whenever a lattice can be represented as

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an interval in a subgroup lattice of a finite group, then it also occurs as a lattice of intermediate subfactors of a von Neumann algebra. With the exception of two lattices, he was able to find intervals isomorphic to every lattice with at most six elements. One of the missing cases was the hexagon lattice. M. Aschbacher [2] gave a general construction whose particular cases provided examples for the hexagon and for the other six-element lattice Watatani was not able to handle. In universal algebra a well-known open problem asks whether every finite lattice is isomorphic to the congruence lattice of a finite algebra. This problem is motivated by the fundamental result of George Gr¨atzer and E. Tam´ as Schmidt [11] stating that every algebraic lattice is isomorphic to the congruence lattice of some algebraic structure. (A lattice is called algebraic iff it is complete and every element is a join of compact elements. In particular, every finite lattice is algebraic.) For almost all finite lattices each known proof of the Gr¨atzer–Schmidt Theorem constructs infinite algebras to represent the lattice as a congruence lattice. So it is a natural question to ask, if finite algebras with prescribed finite congruence lattice can be constructed. In a joint paper with Pavel Pudl´ak [19] we proved that the problem about general algebraic structures is actually equivalent to a group theoretic problem. Problem 3.1 Is every finite lattice isomorphic to an interval in the subgroup lattice of a finite group? One direction of the equivalence is obvious. Let G act on the set of right cosets of the subgroup H, and consider each permutation in G as an operation with one variable. Then the congruences are exactly the partitions into cosets of subgroups belonging to the interval Int(H, G), hence the congruence lattice of this multiunary algebra is isomorphic to this interval. Concerning the reverse implication, it should be emphasized that we do not claim that the congruence lattices of finite algebras are (up to isomorphism) the same as the intervals in subgroup lattices of finite groups. What we proved is that if all finite lattices can be represented as congruence lattices of finite algebras then all finite lattices can be represented as intervals in subgroup lattices of finite groups. In fact, we embed any finite lattice into a finite lattice with some useful properties, and then we show that the smallest algebra with a congruence lattice having these properties is a transitive permutation group considered as a multi-unary algebra. It was shown by Jiˇr´ı T˚ uma [23] that every algebraic lattice is isomorphic to an interval in the subgroup lattice of an infinite group. So it is the finiteness of the group what seems to constitute a severe restriction. Therefore, it is generally believed that the answer to the finite representation problem is negative. Making use of ideas from the fundamental papers of R. Baddeley and A. Lucchini [7], R. Baddeley [6], F. B¨ orner [8], and M. Aschbacher [2] we present here a simplified proof for a slightly modified version of the main result of F. B¨orner [8] giving a reduction of the problem to almost simple groups and to twisted wreath products. Theorem 3.2 Every finite lattice is isomorphic to an interval in the subgroup lattice of a finite group if and only if one of the following is true:

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(1) Every finite lattice consisting of more than one element is isomorphic to an interval Int(H, G) in the subgroup lattice  of an almost simple finite group G with a core-free subgroup H (that is, g∈G g −1 Hg = 1). (2) Every finite lattice consisting of more than one element is isomorphic to an interval Int(D, G) in the subgroup lattice of a twisted wreath product G = Twr(T, D, D0 , ϕ) of a non-abelian finite simple group T and a finite group D with respect to a subgroup D0 < D and a homomorphism ϕ : D0 → Aut(T ) satisfying ϕ(D0 ) ≥ Inn(T ). In the proof we will make use of the following lemmas. Lemma 3.3 Every finite lattice L can be embedded as a filter into a finite lattice ˆ generated by its coatoms (maximal elements in L ˆ \ {1 ˆ }). L L ˆ = L∪{c1x , c2x , ax | x ∈ L\{0L , 1L }}∪{c11 , c12 , c21 , c22 , a1 , a2 , 0∗ } Proof We define L ˆ in the following way: each new element is < 1L ; and we extend the order on L to L ax < y if x ≤ y in L; a1 , a2 < y for all y ∈ L; 0∗ is smaller than every other element ˆ ax < c1x , c2 , a1 < c11 , c12 , a2 < c21 , c22 . It is straightforward to check that L ˆ of L; 2 ˆ consisting of the elements above 0L . Moreover, is a lattice and L is the filter in L the elements c1x , c2x , c11 , c12 , c21 , c22 are coatoms and they generate the whole lattice ˆ since ax = c1 ∧ c2 , a1 = c11 ∧ c12 , a2 = c21 ∧ c22 , 0∗ = a1 ∧ a2 , 0L = a1 ∨ a2 , L, x x x = ax ∨ 0L (x ∈ L \ {0L , 1L }), and 1L = c11 ∨ c12 .  Lemma 3.4 If N  G and N ≤ H, then Int(H, G) ∼ = Int(H/N, G/N ). Lemma 3.5 If N  G, H < G with N H = G, then Int(H, G) ∼ = IntH (H ∩ N, N ). Proof It is easy to check that the maps U → U ∩N (U ∈ Int(H, G)) and V → V H (V ∈ IntH (H ∩N, N )) are order-preserving, and are inverses to each other, showing the isomorphism of the two intervals.  Lemma 3.6 Assume that Int(H, G) is a strongly non-modular lattice. Then for any normal subgroup N  G either N ≤ H or N H = G. Moreover, if H is core-free, then G has a unique minimal normal subgroup M and M is not abelian. Proof Suppose that H < N H < G. Since the interval is strongly non-modular, there exist subgroups H < X < Z < G with N H ∧ X = N H ∧ Z and N H ∨ X = N H ∨ Z. Now N X is a subgroup and N X = N ∨ X = N ∨ H ∨ X = N H ∨ X = N H ∨ Z ≥ Z, hence every z ∈ Z can be written as z = nx with n ∈ N and x ∈ X. Then n = zx−1 ∈ N H ∧ Z = N H ∧ X ≤ X, so z = nx ∈ X, that is Z ≤ X, a contradiction. Suppose now that H is core-free, and let M  G be a minimal normal subgroup. Then M H = G, and by (3.5) Int(H, G) ∼ = IntH (H ∩ M, M ). This lattice is not modular, hence M cannot be abelian. If M ∗ is another minimal normal subgroup, then M and M ∗ elementwise commute, so any H-invariant subgroup of M is also M ∗ H-invariant, i.e., normal in G (since M ∗ H = G). Then the minimality of M

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gives that IntH (H ∩ M, M ) can have at most two elements, although we have included in the definition of strongly non-modular lattices that they have more than two elements.  Proof of the Theorem We have to prove that if both (1) and (2) fail, then there exists a finite lattice that cannot be represented as an interval in the subgroup lattice of a finite group. Let L1 be a lattice that is not isomorphic to Int(H, G) in the subgroup lattice of any finite almost simple group G with a core-free subgroup H, and let L2 be a lattice that is not isomorphic to Int(D, G) in the subgroup lattice of any twisted wreath product G = Twr(T, D, D0 , ϕ) of a non-abelian finite simple group T and a finite group D with respect to a subgroup D0 < D and a homomorphism ϕ : D0 → Aut(T ) satisfying ϕ(D0 ) ≥ Inn(T ). Let us embed L1 as ˆ 1 that is generated by its coatoms (see (3.3)). Now let a filter into a finite lattice L ˆ 1 , L2 , their duals L be the following lattice assembled together using the parts L ˆ d , Ld , and two hexagons: L 1 2 Ld2

ˆ 1 ⊃ L1 L

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Notice that L is a self-dual lattice. It is also easy to see that L is a strongly non-modular lattice. We will show that our assumptions on L1 and L2 imply that L cannot be isomorphic to an interval in the subgroup lattice of any finite group. Suppose the contrary, that there exist a finite group G and a subgroup H < G such that Int(H, G) ∼ = L. We assume that G has minimal order among groups with an interval isomorphic to L in Sub(G). Clearly, H is a core-free subgroup in G (cf. (3.4)). Let M be a minimal normal subgroup of G. Since the interval Int(H, G) is a strongly non-modular lattice, (3.6) implies that M H = G, M is the unique minimal normal subgroup in G, and M = T1 × · · · × Tk (k ≥ 1) with pairwise isomorphic non-abelian simple groups T1 ,. . . ,Tk . Now H permutes the direct factors of M transitively. Let H1 = NH (T1 ) and let α : H1 → Aut(T1 ) give the automorphisms of T1 induced by conjugation by elements of H1 . Furthermore, take coset representatives x1 = 1, x2 , . . . , xk of H1 in −1 H so that x−1 i T1 xi = Ti (i = 1, . . . , k). For each i fix the isomorphism t → xi txi k (t ∈ T1 ) between T1 and Ti , so that M becomes T (where T = T1 ). As in Section 2

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P´alfy: Subgroups of twisted wreath products

we may identify M with Sdp(H1 , α) ≤ T |H| . Now Int(H, G) ∼ = IntH (H ∩ M, M ) (see (3.5)). Step 1. We show that H ∩ M = 1. Following (2.8) we distinguish four cases for the subgroup H ∩ M : 1. H ∩ M is a subdirect product in M = T k ; 2. H ∩M is a box, i.e., H ∩M = U k for some H1 -invariant subgroup 1 < U < T ; 3. H ∩ M is a skew subgroup, i.e., a nontrivial subgroup properly contained in a box; 4. H ∩ M = 1. If H ∩ M is a subdirect product, then every subgroup containing it is also a subdirect product. In virtue of (2.2) let H ∩ M be determined by a subgroup H0 ≤ H containing H1 and a homomorphism β : H0 → Aut(T ) extending α. Then (2.5) implies that IntH (H ∩ M, M ) is isomorphic to the dual of the interval Int(H1 , H0 ), so this latter lattice is also isomorphic to L, contrary the minimal choice of G. If H ∩ M = U k is a box, then it follows from (2.9) and (2.12) that all maximal ˆ1 invariant subgroups in IntH (H ∩ M, M ), i.e., all coatoms are also boxes. Since L is generated by coatoms, we obtain that all invariant subgroups in this subinterval ˆ1 ∼ are boxes, hence L = IntH1 (V, T ) for an appropriate H1 -invariant subgroup V . As ˆ L1 is a filter in L1 , we obtain that L1 occurs as an interval in the subgroup lattice of the almost simple group T H1 /CT H1 (T ), contrary to our assumption on L1 . If H ∩ M is a skew subgroup, then let U k be the box containing it, i.e., the direct product of the images of the projections of H ∩ M . Then H ∩ M < U k < M = T k , and at least one of the hexagons in IntH (H ∩ M, M ), say, Int(X, Z) (where X < Z ∈ IntH (H ∩ M, M )) does not contain U k . The top element of the hexagon, Z is a maximal H-invariant subgroup in T k , hence it is either a subdirect product or a box (see (2.9)). If it were a box, then Z ∩ U k = H ∩ M would be a box as well, which is not the case. So Z is a subdirect product, and then (2.12) implies that X is also a subdirect product. In this case, however, Z ∩ U k = X ∩ U k = H ∩ M = 1 cannot hold by (2.13). We conclude that H ∩ M = 1, so G = M H is a semidirect product, in fact, it is the twisted wreath product Twr(T, H, H1 , α). Step 2. We exclude the possibility that the top elements of both hexagons in IntH (1, M ) are boxes. Assume that these coatoms of L correspond to the subgroups U1k and U2k . Then U1 and U2 are maximal H1 -invariant subgroups of T . Since the normalizer of a H1 invariant subgroup is also H1 -invariant and T is a simple group, it follows that U1 and U2 are self-normalizing. As U1k ∩U2k = H∩M = 1, we get NT (U1 )∩NT (U2 ) = 1. Therefore, if h ∈ H1 induces an inner automorphism on T , then it is the trivial automorphism, that is, α(H1 ) ∩ Inn(T ) = 1. We distinguish two cases, whether α(H1 ) = 1 or not, and show that in both cases at least one of the subgroups U1 , U2 is a p-group for some prime number p. If α(H1 ) = 1, then all subgroups of T are H1 -invariant. Now U1 is a p-group,

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465

otherwise two Sylow subgroups corresponding to different prime divisors of |U1 | would provide two H-invariant subgroups of U1k with trivial intersection, which is not the case as U1k is the top element of one of the hexagons in L. Now consider the case when α(H1 ) is non-trivial. We have seen that α(H1 ) ∩ Inn(T ) = 1, hence α(H1 ) is isomorphic to a subgroup of Out(T ), which is a solvable group according to Schreier’s Hypothesis. Let A be a minimal normal subgroup in α(H1 ) ≤ Aut(T ). This is an elementary abelian q-group for some prime q. Now CT (A) < T is a proper H1 -invariant subgroup. (It may be the trivial subgroup.) We can choose one of i ∈ {1, 2} so that the H-invariant box subgroup CT (A)k intersects Uik trivially. Then we have CT (A) ∩ Ui = 1, that is, A acts fixed point freely on Ui . It follows that q does not divide |Ui |, and for every prime divisor p of |Ui | there is a unique A-invariant Sylow p-subgroup of T (see [1, 18.7]). The uniqueness implies that this Sylow subgroup is also H1 -invariant. It follows, as before, that Ui is a p-group. We have shown that in both cases Ui (i = 1 or 2) is a p-group. Observe that the Frattini subgroup Φ(Uik ) is also H-invariant, as it is a characteristic subgroup of Uik . Since Uik /Φ(Uik ) is abelian, the interval IntH (Φ(Uik ), Uik ) is a modular lattice consisting of at least two elements. Moreover, by the basic property of the Frattini subgroup, if W ∨Φ(Uik ) = Uik for some subgroup W , then W = Uik . However, there is no element in the lattice L that has these properties required from Φ(Uik ). This contradiction shows that it is not possible that the top elements of both hexagons are boxes. Step 3. Conclusion of the proof. Now we may assume that the top element of one of the hexagons in IntH (1, M ) ∼ = L is a subdirect product. Let this hexagon be Int(X, Z). By (2.12) we see that X is also a subdirect product. Hence (2.2) and (2.3) yield that X = Sdp(HX , β) for a subgroup HX ≤ H and a homomorphism β : HX → Aut(T ), and 9 the subgroups of M containing X are exactly the subdirect products Sdp(W, β 9W ) correspond9 ing to subgroups W ∈ Int(H1 , HX ). (In particular, we have α = β 9H1 .) Let 9 Z = Sdp(HZ , β 9H ) ∼ = T |H:HZ | . Since IntH (1, Z) does not contain any box or Z skew subgroup, we must have β(HZ ) ≥ Inn(T ) (see (2.11)). Let K be the kernel of β. Since Int(HZ , HX ) is a hexagon, and that is a strongly non-modular lattice, it follows from (3.6) that either K ≤ HZ or KHZ = HX . In the first case Inn(T ) ≤ β(HZ ) ∼ = HZ /K < HX /K ∼ = β(HX ) ≤ Aut(T ), so the hexagon Int(HZ , HX ) ∼ = Int(β(HZ )/Inn(T ), β(HX )/Inn(T )) is an interval in the subgroup lattice of the outer automorphism group of T . However, the outer automorphism group is solvable by Schreier’s Hypothesis, but the subgroup lattice of a solvable group cannot contain a hexagon as an interval by (3.6). Thus we have KHZ = HX , and so KH1 = HX as well, thus β(H1 ) = β(HZ ) ≥ Inn(T ). By (2.4) this means that with the exception of the trivial subgroup, IntH (1, M ) consists of subdirect products only. In particular, all elements apart from the trivial subgroup in the interval corresponding to L2 are subdirect products, hence L2 occurs as an interval in the subgroup lattice of a twisted wreath product as in (2), contrary to our assumption. This finishes the proof of Theorem 3.2. 

466

4

P´alfy: Subgroups of twisted wreath products

Comments on related literature

The definition of the twisted wreath product is given in the books by Bertram Huppert [13] and by Michio Suzuki [22]. Huppert [13, Definition I.15.10] gives the same definition as in the original paper of B. H. Neumann [17], and uses the German name verschr¨ anktes Kranzprodukt. Suzuki’s definition [22, Chapter 2, Definition 10.3] is essentially the same as we have formulated it in Section 2. It is worth mentioning the analogy between twisted wreath products and induced representations, noticed by Dan Haran [12, Section 1]. The problem of representing finite lattices as intervals in subgroup lattices has raised considerable interest. For a survey see [18]. Although a negative answer is expected, only some deep reduction theorems and solutions for particular classes of lattices have been achieved so far. Recently Michael Aschbacher devoted several voluminous works to this problem. Here we can mention only two of these: one dealing with overgroups of root subgroups in classical groups [4], another investigating intervals in the subgroup lattice of alternating and symmetric groups [3]. John Shareshian [21] suggested some candidates for lattices that may not be representable as intervals in subgroup lattices of finite groups. William DeMeo [10] found representations of all lattices consisting of at most 7 elements, with two exceptions:

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So currently these are the smallest lattices for which no representation as an interval in the subgroup lattice of a finite group is known. In the present paper our goal was to combine the ideas of Aschbacher, Baddeley, B¨orner, and Lucchini ([2], [6], [7], [8]) in order to give an accessible proof of the reduction theorem (3.2). The statement of the theorem sligthly differs from B¨orner’s version. On one hand, we have improved case (1) by stating it for all lattices not just for those generated by coatoms. This was made possible by the embedding lemma (3.3). (B¨orner [8, Lemma 1.1] found a less useful embedding.) On the other hand, our version of (2) is slightly weaker than his, we do not get that D0 is a core-free subgroup of D. That was achieved by B¨orner with the help of a more complex lattice than our L and using some additional arguments. The main ideas are also present in the other papers. However, Baddeley [6] gives a reduction that can somewhat alter the lattice to be represented, and Aschbacher [2] formulates the alternative only for a special class of lattices what he calls CD-lattices. Baddeley and Lucchini [7] study lattices of height 2. Finally, let us point out some substantial parts from these papers that were used in our presentation.

P´alfy: Subgroups of twisted wreath products

467

• (2.5) and (2.6): [2, (7.1)]. On the basis of this remark Aschbacher concentrates on the kernels of the extensions of the homomorphism ϕ : D0 → Aut(T ). These are subgroups normalized by D0 and together with an additional top element form what he calls (the dual of) a signalizer lattice. • (2.7): Aschbacher’s example for the hexagon [2, Example 8.5] is slightly different, he takes T = A5 , D = A6 × A6 , D0 = diag(A5 ). • (2.13): [8, Lemma 4.10] • (3.6): Baddeley [6, Definition 3.3] uses a weaker condition, what he calls QP-property. In contrast, B¨orner’s LP-property [8, Definition 2.1] is stronger than ours, making the unnecessary requirement that y∧x = 0L and y∨x = 1L . Aschbacher’s A-lattices [2, p. 810] are exactly those what we call strongly non-modular. The conclusion of (3.6) is obtained in each of these papers, and it is emphasized that this corresponds to the notion of quasiprimitive permutation groups. • Step 1 in the proof of (3.2): [8, Theorems 5.1, 5.2, 5.3] • Step 2 in the proof of (3.2): [8, Theorem 5.6], [6, part of Theorem 4.9] • Step 3 in the proof of (3.2): [8, Lemma 5.4] Acknowledgement The author has been supported by the National Research, Development and Innovation Fund of Hungary, grant no. 115799. References [1] M. Aschbacher, Finite Group Theory, Cambridge University Press, 1986. [2] M. Aschbacher, On intervals in subgroup lattices of finite groups, J. Amer. Math. Soc. 21 (2008), 809–830. [3] M. Aschbacher, Lower signalizer lattices in alternating and symmetric groups, J. Group Theory 15 (2012), 151–225. [4] M. Aschbacher, Overgroups of root groups in classical groups, Mem. Amer. Math. Soc. 241 (2016), no. 1140. [5] M. Aschbacher and L. L. Scott, Maximal subgroups of finite groups, J. Algebra 92 (1985), 44–80. [6] R. Baddeley, A new approach to the finite lattice representation problem, Period. Math. Hungar. 36 (1998), 17–59. [7] R. Baddeley and A. Lucchini, On representing finite lattices as intervals in subgroup lattices of finite groups, J. Algebra 196 (1997), 1–100. [8] F. B¨orner, A remark on the finite lattice representation problem, in Contributions to General Algebra, 11 (I. Chajda et al., eds.) (Verlag Johannes Heyn, Klagenfurt, 1999), 5–38. [9] P. J. Cameron, Permutation Groups, London Math. Soc. Student Texts 45, Cambridge University Press, 1999. [10] W. DeMeo, Congruence lattices of finite algebras, (PhD Thesis, University of Hawaii, 2012) arXiv:1204.4305 [11] G. Gr¨ atzer and E. T. Schmidt, Characterizations of congruence lattices of abstract algebras, Acta Sci. Math. (Szeged) 24 (1963), 34–59. [12] D. Haran, Hilbertian fields under separable algebraic extensions, Invent. Math. 137 (1999), 113–126.

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[13] B. Huppert, Endliche Gruppen I, Grundlehren der mathematischen Wissenschaften 134 (Springer-Verlag, Berlin–Heidelberg, 1967) [14] L. G. Kov´ acs, Maximal subgroups in composite finite groups, J. Algebra 99 (1986), 114–131. [15] J. Lafuente, On restricted twisted wreath products of groups, Arch. Math. (Basel) 43 (1984), 208–209. [16] M. W. Liebeck, C. E. Praeger and J. Saxl, On the O’Nan–Scott theorem for finite primitive permutation groups, J. Austral. Math. Soc. Ser. A 44 (1988), 389–396. [17] B. H. Neumann, Twisted wreath product of groups, Arch. Math. (Basel) 14 (1963), 1–6. [18] P. P. P´alfy, Intervals in subgroup lattices of finite groups, in Groups ’93 Galway/St Andrews, Vol. 2 (C. M. Campbell, T. C. Hurley, E. F. Robertson, S. J. Tobin, J. J. Ward, eds.) London Math. Soc. Lecture Notes Ser. 212 (Cambridge University Press, 1995), 482–494. [19] P. P. P´ alfy and P. Pudl´ak, Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups, Algebra Universalis 11 (1980), 22–27. [20] L. L. Scott, Representations in characteristic p, in Santa Cruz Conference on Finite Groups, Proc. Sympos. Pure Math. 37 (Amer. Math. Soc., Providence, R.I., 1980), 318–331. [21] J. Shareshian, Topology of order complexes of intervals in subgroup lattices, J. Algebra 268 (2003), 677–686. [22] M. Suzuki, Group Theory I, Grundlehren der mathematischen Wissenschaften 247 (Springer-Verlag, Berlin–Heidelberg–New York, 1982). [23] J. T˚ uma, Intervals in subgroup lattices of infinite groups, J. Algebra 125 (1989), 367–399. [24] Y. Watatani, Lattices of intermediate subfactors, J. Funct. Anal. 140 (1996), 312–334.

SOME REMARKS ON SELF-DUAL CODES INVARIANT UNDER ALMOST SIMPLE PERMUTATION GROUPS B. G. RODRIGUES and T. M. MUDZIIRI SHUMBA School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban 4000, South Africa Emails: [email protected], [email protected]

Abstract We present some recent advances on the question of enumeration and classification of self-dual codes invariant under finite non-solvable permutation groups. In particular, we examine the question for almost simple groups of sporadic or rank 3 types. As a by-product we determine all doubly even and extremal codes of length n ≤ 4095 which admit an almost simple group of sporadic or rank 3 type as a permutation group of automorphisms acting transitively on the code’s coordinate positions.

1

Introduction

The question of existence of self-dual codes of moderate lengths has attracted the attention of researchers in coding theory for several years now. Many classification and enumeration results can be found in recent literature. In particular, and for length n ≤ 72, in [15] W. Cary Huffman gave a survey of what was known at the time on the classification and enumeration of self-dual codes of moderate lengths. In order to find examples for larger n a non-trivial automorphism group might be useful. Thus an approach that is often used to address the problem of whether a self-dual code C of a given length exists is to assume the invariance of C under some nontrivial automorphism and then try either to construct such a code or to prove its non-existence. In [5] Chigira, Harada and Kitazume constructed a binary self-dual code with parameters [100, 50, 10] with automorphism group the group J2 :2 the extension of the Hall-Janko group J2 by its involutory outer automorphism. This code is spanned by its codewords of weight 14, which are described as the set of fixed points of the outer involutions. Similarly, the extended Golay code G24 is spanned by the codewords of weight 8, namely the octads, and these form the set of fixed points on involutions in class 2A. Based on these observations, and under the assumption that the group G of the code is a sporadic almost simple group, a number of selfdual codes of length n ≤ 1000 were determined by Chigira, Harada and Kitazume in [4]. In that paper, the authors gave a construction of a code C(G, Ω) as the dual of a code spanned by fixed points of involutions of a permutation group G on a set Ω. In fact, they showed that any self-dual code C satisfies C(G, Ω)⊥ ⊆ C ⊆ C(G, Ω). Recently, in [22], Mukwembi, Rodrigues and Shumba extended the results of [4] to length n ≤ 4095 and determined a large number of self-dual codes (see Tables 1

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Rodrigues, Shumba: Almost simple groups and self-dual codes

to 7) admitting an almost simple sporadic group as a permutation automorphism group. In that paper, a considerable number of self-dual codes are presented. In a related study, under the assumption that G acts 2-transitively on the coordinate positions of an extremal code, Malevich and Willems [19] gave a classification of extremal self-dual doubly even codes C invariant under G, stopping short of showing the non-existence of a putative extremal self-dual doubly even code of length 1024 invariant under the group E  SL(2, 25 ), where E is an elementary abelian group of order 1024. In [6] Chigira, Harada and Kitazume completed the characterisation by showing that there does not exist an extremal self-dual doubly even code of length 1024 which admits the group E  SL(2, 25 ). Following the ideas of [19], Rodrigues (see [26] and [28]) examined the question of existence (respectively non-existence) of self-dual codes invariant under a rank 3 group of almost simple type and produced a number of examples of self-dual codes that were not found in [5] or [22]. In this paper, we report on the progress obtained thus far on the problem concerning enumeration and classification to the extent where possible of all self-dual codes of length n invariant under certain types of groups, in particular almost simple groups of sporadic or rank 3 types. In essence, this paper reports on answers obtained to the following group theoretical question: Can we determine, up to isomorphism, the number of self-dual codes of length n on which a prescribed almost simple group G of sporadic or rank 3 type acts transitively as a permutation automorphism group? Due to computer time limitations and in view of the current knowledge on the classification of primitive groups we restrict our attention to lengths n ≤ 4095.

2

Preliminaries and Notation

Let κ = F2 denote the binary field and G be a finite group. As usual κG denotes the group  algebra consisting of κ-linear combinations of elements of G, namely, κG = { g∈G ag g|ag ∈ κ, g ∈ G}. A linear code C can then be regarded as a Gsubmodule of κG. Given a linear code C, the dual code C ⊥ is defined as C ⊥ = {u ∈ C | u.v = 0 for all v ∈ C}, where . is the classical inner product. A linear code C is called self-orthogonal if C ⊆ C ⊥ . If C = C ⊥ we say that C is self-dual. The support of u = (u1 , . . . , un ) ∈ C is defined as {i ∈ {1, . . . , n}|ui = 0}. The weight of u denoted wt(u) is the cardinality of the support. If C is a self-dual code, then clearly wt(u) ≡ 0 (mod 2) holds for all u ∈ C. We say a code is even if all codewords have weight divisible by 2 and doubly even if all codewords have weight divisible by 4. A doubly even self-dual code is also called a Type II code. If a code is even but not doubly even, then we say that it is singly even or Type I. A Type II code of length n exists if and only if n ≡ 0 (mod 8). Two linear codes of the same length and over the same field are equivalent if each can be obtained from the other by permuting the coordinate positions and multiplying each coordinate position by a non-zero field element. They are isomorphic if they can be obtained from one another by permuting the coordinate positions. Note that for binary codes, the

Rodrigues, Shumba: Almost simple groups and self-dual codes

471

two notions are equivalent and henceforth we will refer to code inequivalence. An automorphism of a code C is any permutation of the coordinate positions that maps codewords to codewords. The automorphism group will be denoted by Aut(C). We note that our point of divergence from the notation used in [4] is that instead of working with a set of fixed points of involutions, we turn the set into a vector via the map P(Ω) → κn2 , A → v = (v1 , . . . , vn ) where vi = 1 or 0, according to whether i ∈ A or not, and P(Ω) is the power set of the n-point set Ω. Then the inner product (A, B) ≡ |A ∩ B| mod 2 becomes the classical inner product.

3

Main theorems and examples

Using ideas similar to those in [4], we view codes as G-submodules and find all G-submodules satisfying C(G, Ω)⊥ ⊆ C ⊆ C(G, Ω). Our computations are carried out with the help of the computational Algebra systems Magma [1, 2] and Gap [12]. In order to check whether a group G occurs as an automorphism group of a selfdual code C of length n, we exploit the structure of C as a submodule of the space κG as a G-module. The code C is a submodule of dimension n2 in κG. A necessary and sufficient condition for existence of a self-dual doubly even code is given in [14, Theorem 5.2]. The following result on automorphism groups of codes is a theorem in [22]. Theorem 3.1 Let G be a finite sporadic simple group or sporadic almost simple group of degree n. Suppose that G acts transitively on a set Ω of size n ≤ 4095. Then the following hold: (i) If G acts primitively on Ω and either (a) Out(G) = 1 or (b) Out(G) = 2, then in case (a) Aut(C(G, Ω)) ∼ = G and in case (b) Aut(C(G, Ω)) ∼ = G:Out(G) = G:2. The only exceptions to this, namely (i)(a) and (i)(b) are G = M11 with n = 11, 12, 55; G = M12 with n = 12, 66, and G = J1 with n = 266 for which Aut(C(G, Ω)) = Sn , save for M12 , n = 66 which gives Aut(C(M12 , 66)) = S12 . For G = M22 , n = 176, both M22 and HS act primitively on the set of points and C(G, 176) is equivalent to C(HS, 176) and Aut(C(M22 , 176)) = HS. (ii) If G acts imprimitively on Ω, with |Ω| = n even, but primitively on Ω , with |Ω | = n2 , and additionally: n (a) If Out(G) = 1, then Aut(C(G, Ω)) ∼ = 2 2  G = Z2  G. (b) If Out(G) = 2 and Aut(G) ∼ = G:2 acts primitively on Ω , then n 2 Aut(C(G, Ω)) ∼  G:2 = Z2  G:2. 2 =

The only exceptions to this are G = M11 with n = 22, 110, 330; G = M12 with n = 24, 132 and furthermore, for all exceptions, save M12 , n = 132, Aut(C(G, Ω)) ∼ =

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Rodrigues, Shumba: Almost simple groups and self-dual codes

n

2 2  Aut(C(G, Ω )) = Z2  S n2 . For M12 , n = 132, Aut(C(M12 , 66)) = S12 and Aut(C(M12 , 132)) = 266 :S12 . 3.1

Codes invariant under almost simple sporadic groups

Tables 1 to 7 contain some previously known self-dual codes invariant under sporadic or almost simple sporadic groups (see [4]) with parameters [24, 12, 8] for G = M12 :2, M24 , [22, 11, 6] for G = M22 , M22 :2, [100, 50, 10] for G = J2 :2, [132, 66, 6], [132, 66, 12] (two inequivalent codes with these parameters) for G = M11 , [132, 66, 12] (three inequivalent codes with these parameters) for G = M12 :2, [220, 110, 18], [220, 110, 20] (two inequivalent codes with these parameters) for M12 , [330, 165, 10] for G = M22 :2 and [352, 176, 16] (three inequivalent codes with these parameters) for G = HS:2. Remark 3.2 The reader would have noticed that [4, Section 3] provides some classification results for self-dual codes of lengths n ≤ 1000 which are invariant under certain simple and almost simple groups of sporadic type. In the sequel, in an attempt to give an answer to the question posed above, we confirm and extend the results of [4] and hence give new classification and enumeration results on inequivalent self-dual codes of lengths up to 4095 invariant under the said type of groups. According to [4, Lemma 2.11], there are possibilities of existence of new G-invariant self-dual codes in the following cases (see results listed in Tables 1 to 7 below): G = M11 of degree of degree 22, 110, 132, 330, 440, 660 and 792. G = M12 of degree 132, 220, 264, 440, and 792. G = M12 :2 of degree 132, 264, 288, 440, 792 and 990. G = M22 of degree 330, 462, and 1232. G = M22 :2 of degree 44, 154, 330, and 462. G = M23 of degree 506. G = M24 of degree 562. G = HS of degree 352. G = HS:2 of degree 200, 352, and 704. G = J2 :2 of degree 200, 560, 630, and 1050. G = Co3 of degree 552. G = Ru of degree 4060. Our notation for groups is standard and follows that of the ATLAS [7]. We use permutation representations of a given group from the GAP table of marks to get generators of the group as well as its subgroups. In some cases complete classification was possible but in many instances we could only determine the number of self-dual codes invariant under a prescribed permutation group but no information on the (non)equivalence or (non)isomorphism of the codes of a particular length could be obtained with the computational resources available. In the ensuing tables, by Rep we refer to the representation index in the GAP table of marks, n is

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473

the cardinality of the G-set Ω = G/H for some H ≤ G, δ(C) is the quantity of self-dual codes of length n invariant under the group “Gp” and  is placed where information could not be obtained about minimum weights and equivalence. In instances where information about minimum weights and isomorphism could not be determined we use the symbol  under the remarks column. In some of the cases we have [n, k, d] (α) under the remarks column to signify that there are α codes with the parameters [n, k, d]. Furthermore, we use the symbol † where we have used some theoretical results given in the paper, for example Theorem 3.1, to determine the automorphism groups of given codes.

Table 1: M11 -invariant self-dual codes.

Gp M11

n 12 22 66 110

132

144 220

330 396 440

660 720 792

Rep 37 36 34 33 32

δ(C) 0 1 0 1 1

31

3

30

3

29

3

28 26

0 9

25

27

24 23 22 21 20

27 3 1 0 15

19 17 16 15 14

 0 567 0 

Remarks [22, 11, 2]-code, Aut 211 :S11 . [110, 55, 2]-code. [110, 55, 2]-code. inequiv, Aut 255 :S55 for both. [110, 55, 6] (2), [110, 55, 2], inequiv, Aut 255 :S55 . [132, 66, 12] (2) inequiv. [132, 66, 6], †. All codes have Aut 266 :M11 . [132, 66, 4] (2) inequiv, [132, 66, 2] (1). For all codes, Aut 266 :M11 . [220, 110, 20] (4), [220, 110, 18] (2) [220, 110, 2]. [220, 110, 4] (14), [220, 110, 10] (2) [220, 110, 12] (10), [220, 110, 2] (1). [220, 110, 4] (24), [220, 110, 2] (3). [330, 165, 6] (1), [330, 165, 8] (2), †, inequiv. Aut 2165 :M11 [330, 165, 2] Aut 2165 :M11 . [440, 220, 8] (12), [440, 220, 4] (2), [440, 220, 2].   

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Table 2: M12 -invariant self-dual codes.

Gp M12

n 12 66 132

144 220 264

396 440

660

792

880 990 1320 1584 1728 1760 1980

2376 2970

Rep 146 145 144 143 142 141 140

δ(C) 0 0 0 0 3 3 9

139 138

3 9

137 136 135 134 133 132

3 0 0 3 3 183

131

183

130 129

0 11

128

11

125 124 123 122 121

0 0 0 0 243

120 119

735 243

118 117 116 113 105 101 100 97 96 95 92 86

0 5 5 0 0 0 0 0 0 0 

Remarks

[132, 66, 2] [132, 66, 2] [132, 66, 8] [132, 66, 2] [132, 66, 2] [132, 66, 8] [132, 66, 2] [132, 66, 2]

(1), (1), (4), (1), (1), (4), (1), (1),

[132, 66, 4] [132, 66, 4] [132, 66, 6] [132, 66, 4] [132, 66, 4] [132, 66, 6] [132, 66, 4] [132, 66, 4]

(2), (2), (2), (2), (2), (2), (2), (2),

†, inequiv, Aut 266 :S12 . same remarks as above. same remarks as above. same remarks as above. same remarks as above. same remarks as above. same remarks as above. same remarks as above.

[220, 110, 18] (1), [220, 110, 20] (2), † inequiv,Aut M12 . [220, 110, 18] (1), [220, 110, 20] (2), same remarks as above. [264, 132, 8] (96), [264, 132, 6] (24) [264, 132, 4] (60), [264, 132, 2] (3). [264, 132, 8] (96), [264, 132, 6] (24) [264, 132, 4] (60), [264, 132, 2] (3). [440, 220, 8] (8), [440, 220, 4] (2) [440, 220, 2] [440,220,8] (8),[440,220,4] (2) [440, 220, 2]

[792, 396, 8] [792, 396, 4]  [792, 396, 8] [792, 396, 4]

(238), [972, 396, 6] (2), †, Aut 2396 :M12 :2. (2), [792, 396, 2], †, inequiv, Aut 2396 :M12 :2. (238), [972, 396, 6] (2), † inequiv, Aut 2396 :M12 :2. (2), [792, 396, 2] all inequiv, Aut 2396 :M12 :2.

, inequiv, Aut 2495 :(M12 :2). , inequiv, Aut 2495 :(M12 :2).

Rodrigues, Shumba: Almost simple groups and self-dual codes Table 3: M12 :2-invariant codes.

Gp M12 :2

n 24 132 144 264

288 396 440 792 880 990

1320 1584 1728 1980

Rep 211 210 209 208 207

δ(C) 1 3 0 0 27

206 205

187 27

204 203 202 201 196 195 194 192 189 188 187 185 183 178 174

57 267 0 35 327 615 0 299 439 439 299 0 555 0 0

Remarks d = 8, Aut M24 , G. [132, 66, 12] (3), Aut M12 :2, inequiv.

[264, 132, 16] (6), [264, 132, 8] (12) [264, 132, 6] (2), [264, 132, 4] (6) [264, 132, 2]  [264, 132, 16] (6), [264, 132, 8] (12) [264, 132, 6] (2), [264, 132, 4] (6) [264, 132, 2] , inequiv, all have Aut 2144 :M12 :2. , inequiv, all have Aut 2144 :M12 :2.  , †, Aut 2396 :M12 :2. Inequiv. . Same remarks as above. , †, , †, , †, , †,

Aut Aut Aut Aut

2495 :M12 :2. 2495 :M12 :2. 2495 :M12 :2. 2495 :M12 :2.

Inequiv. Inequiv. Inequiv. Inequiv.

, †, Aut M12 :2, primitive.

Table 4: M22 -invariant self-dual codes.

Gp M22

n 22 176

616 672 770 1232 1386 22 44

Rep 155 153 152 150 149 148 147 146 145 144 140 137 488 487

δ(C) 1 0 0 1 0 83 171 0 0 0 549 0 1 6

154

485

6

330

483

1

330 462

M22 :2

Remarks d = 6, Aut M22 :2.

d = 10, Aut M22 :2. , †, Aut 2231 :M22 :2. , †, Aut 2231 :M22 :2.

, †, Aut 2616 :M22 :2. Inequiv. d = 6, Aut M22 :2. [44, 22, 2] (1), [44, 22, 4] (2), inequiv, Aut 222 :M22 :2. [44, 22, 6] (1), [44, 22, 8] (2), inequiv, Aut 222 :M22 :2. [154, 77, 2] (1), [154, 77, 8] (1), †, inequiv, Aut 277 :M22 :2. [154, 77, 10] (4), †, inequiv, Aut 277 :M22 :2 for all. d = 10, Aut M22 :2

475

476

Rodrigues, Shumba: Almost simple groups and self-dual codes Table 5: M22 :2-invariant self-dual codes continued.

Gp

n 352

462

616 672 770 1386

Rep 482 481 480

δ(C) 10 0 53

479 478 477 476 474 473 462

106 55 99 0 0 0 0

Remarks [352, 176, 16] (10) [462, 231, 10] (22), [462, 231, 6] (3), †, inequiv, Aut 2231 :(M22 :2) [462, 231, 14] (27), [462, 231, 2], inequiv, Aut 2231 :(M22 :2) , †, inequiv, Aut 2231 :(M22 :2) , †, inequiv, Aut 2231 :(M22 :2) , †, inequiv, Aut 2231 :(M22 :2)

Table 6: M23 , M24 , J1 , HS and HS:2-invariant codes

Gp M23

M24

J1

HS

HS:2

n 506

Rep 200 199

δ(C) 0 16

1288 24 276 552

198 1528 1527 1526

0 1 0 20

1288 266 1540 1596 100 176 176 352

1524 39 36 35 588 587 586 585

0 0 0 0 0 0 0 7

352

584

7

1100 1100 2 100 200

583 582 533 2056 532

0 0 1 0 13

352 704

531 530

3 31

1100 1100

2055 2054

0 0

Remarks [506, 253, 16] (6), [506, 253, 14], †, Aut 2253 :M23 , inequiv. [506, 253, 8] (4), [506, 253, 6](5), †, inequiv, Aut 2253 :M23 . G24 . [552, 276, 16] (11), [552, 276, 14], Aut 2276 :M24 , so inequiv. [552, 276, 8] (4), [552, 276, 6], †, Aut 2276 :M24 so inequiv. [552, 276, 4] (2), [552, 276, 2], †, Aut 2276 :M24 , inequiv.

[352, 176, 12] (4), [352, 176, 4] (2), †, inequiv, Aut 2176 :HS. [352, 176, 2], †, inequiv, Aut 2176 : HS [352, 176, 12] (4), [352, 176, 4] (2) [352, 176, 2], †, inequiv, Aut 2176 : HS

[2, 1, 2], Aut S2 [200, 100, 4](2), [200, 100, 2](1), †, Aut 2100 :(HS:2), inequiv. [200,100,12](6),[200,100,16](4), †, inequiv, Aut 2100 :(HS:2). [352, 176, 16] (3) [704, 352, 16](6), [704, 352, 12](16) [704, 352, 8](2), [704, 352, 4](6) [704, 352, 2]

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Table 7: Self-dual codes invariant under Co3 , J2 and J2 :2

Gp Co3

J2

n 276 552 100

Rep 2482 2481 145

δ(C) 0 5 3

280 560 840 1008 1050 1800 1890 100 200

144 141 140 139 138 135 134 371 370

0 0 0 0 177 0 0 1 23

280

369 366 365

0 165 295

, †, Aut 2280 :(J2 :2), inequiv , †, Aut 2280 :(J2 :2), inequiv

364

201

, †, Aut 2280 :(J2 :2), inequiv

363 362 361 360 359 358 353 352

25 0 0 61 25 25 0 0

, †, Aut 2315 :(J2 :2), inequiv

560

J2 :2

630 840 1008 1050 1800 1890

Remarks [552, 276, 2] (2), [552, 276, 4], [552, 276, 12](2) Aut 2276 : Co3 , inequiv [100, 50, 16] (2), Inequiv, Aut J2 , [100, 50, 10] (1) [100, 50, 10] code is the same as the one from J2 :2.

[1050, 525, 10] (176), [1050, 525, 2]. Aut 2525 :J2 :2. inequiv.

a [100, 50, 10]-code, Aut J2 :2 [200, 100, 16] (19), [200, 100, 10] [200, 100, 4] (2), [200, 100, 2], †, inequiv, Aut 2100 :(J2 :2)

[1050, 525, 10](60), [1050, 525, 2], †, Aut 2525 :(J2 :2), inequiv. [1050, 525, 10](24), [1050, 525, 2], †, Aut 2525 :(J2 :2), inequiv. [1050, 525, 10](24), [1050, 525, 2], †, Aut 2525 :(J2 :2), inequiv.

Now, using Remark 3.2 we collect the results of Tables 1 to 7 in the following: Theorem 3.3 Let G be one of the sporadic simple and sporadic almost simple groups and suppose that G is transitive on a set Ω of size n, with n ≤ 4095. Let l be the representation index of the table of marks of G. Then C is a self-dual code of length n invariant under G if and only if n, l and G satisfy the following statements: (a) For G = M11 , (n, l) = (22, 36), (110, 31; 32; 33), (132, 29; 30), (220, 24; 25; 26), (330, 22; 23), (440, 19; 20), (660, 16), (792, 14). (b) For G = M12 , (n, l) = (132, 137; 138; 139; 140; 141; 142), (220, 133; 134), (264, 131; 132), (440, 128; 129), (792, 119; 120; 121), (990, 116; 117). (c) For G = M12 :2, (n, l) = (24, 211), (132, 210), (264, 205; 206, 207), (288, 203; 204), (440, 201), (792, 195; 196), (990, 187; 188; 189; 192), (1584, 183). (d) For G = M22 , (n, l) = (22, 155), (330, 150), (462, 147; 148), (1232, 140). (e) For G = M22 :2, (n, l) = (22, 488), (44, 487), (154, 485), (330, 483), (462, 477; 478; 479; 480).

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(f ) For G = J2 , (n, l) = (100, 145), (1050, 138) (g) For G = J2 :2, (n, l) = (100, 371), (200, 370), (560, 364; 365; 366), (630, 363), (1050, 358; 359; 360) (h) For G = M23 , (n, l) = (506, 199). (i) For G = M24 , (n, l) = (24, 1528), (562, 1526). (j) For G = HS, (n, l) = (352, 584; 585). (k) For G = HS:2, (n, l) = (200, 532), (352, 531), (704, 530). (l) For G = Ru, n = 4060. (m) For G = Co3 , n = 552. The number δ(C) of self-dual codes invariant under G is known for each case in (a) – (m). 3.2

Examples

Here, we give a sample of results obtained in [22]. For a complete description of results on the enumeration and classification of self-dual codes that admit almost simple sporadic groups, the reader is encouraged to consult [22]. Proposition 3.4 Let G = M11 and |Ω| = 132. Then, up to isomorphism, there are 6 self-dual codes of length 132 invariant under G. Further, the automorphism group for each of these codes is 266 :M11 Proof Let G = M11 and |Ω| = 132. By the GAP table of marks [12], M11 has two inequivalent imprimitive permutation representations of degree 132, namely those of indices 30 and 29 in the list of subgroups (see lines 7 - 8 of Table 1). Let Ω1 and Ω2 be the corresponding coset spaces for the groups which are the images of the permutation action of G on 132 points. Then by Magma [1], we have that C(G, Ω1 ) is a [132, 67, 6]-code and C(G, Ω2 ) is a [132, 67, 2]-code. There are three self-dual codes between C(G, Ω1 )⊥ and C(G, Ω1 ). We denote these C1 , C2 and C3 , respectively. By MAGMA, the codes C1 and C2 , have minimum weight 12 and C3 has minimum weight 6. These codes have been shown to be inequivalent in [4, Example 3.6]. We now show that there are three new inequivalent self-dual codes between C(G, Ω2 )⊥ and C(G, Ω2 ), say D1 , D2 and D3 . Two of these, D1 and D2 , have minimum weight 4 and D3 has minimum weight 2. The group G acts on the set {D1 , D2 , D3 } of codes Di . By the Orbit-Stabilizer Theorem, we have [G : GDi ] ≤ 3, using the fact that the set {D1 , D2 , D3 } has three elements. But G has no subgroups of index ≤ 3, (see [9]). It follows that Di , 1 ≤ i ≤ 3 is G-invariant. Now, we determine the automorphism group Aut(Di ). Since for each i where 1 ≤ i ≤ 3 we have Di is G-invariant. Further, Out(G) = 1 and G acts primitively on 66 points. By Result 3.1, since 132 = 2(66), we have Aut(C(G, 132)) = 266 :Aut(C(G, 66)) ∼ = 266 :M11 . Each of the codes Di is 266 :M11 66 invariant from which we have 2 :M11 ≤ Aut(Di ) ≤ Aut(C(G, 132)) = 266 :M11 whence Aut(Di ) = 266 :M11 . Since these codes are distinct and have the same automorphism group, they are inequivalent by [4, Lemma 2.3]. 2 Proposition 3.5 Let G = M12 . Then the following hold.

Rodrigues, Shumba: Almost simple groups and self-dual codes

479

(i) Up to isomorphism, there are 30 self dual codes of length 132 invariant under G. (ii) If |Ω| = 220, then there are 6 self-dual codes invariant under G, up to equivalence, of this length. (iii) If |Ω| = 264, then there are 366 self-dual codes invariant under G, (not necessarily inequivalent), of this length. Further, of these codes, 108 are doubly even with 60 having parameters [264, 132, 4] and 48 having parameters [264, 132, 8]. (iv) There are 22 self-dual binary codes invariant under G of length 440. Of these, 16 are doubly even with 12 such doubly even codes having parameters [440, 220, 8] and 4 having parameters [440, 220, 4]. (v) There are 1221 self-dual binary codes of length 792 invariant under G, up to isomorphism. Of these, 442 are doubly even. 238 coming from the first representation, 238 from the second representation and 102 from the third representation. (vi) Up to isomorphism, there are 8 self-dual binary codes of length 990 invariant under G. Proof (i) Here G acts primitively on 66 = 132 2 points, Aut(C(G, 66)) = S12 hence Aut(C(G, 132)) = 266 :S12 . This group acts on the coordinate spaces of all the 30 self-dual codes and we conclude that it is the automorphism group of each of them and hence that they are inequivalent. (ii) Let G = M12 and |Ω| = 220. From Table 2 (see lines 13 - 14), there are three self-dual codes of length 220 for both representations. By Theorem 3.1, Aut(C(G, 220)) = G and this acts on each of the codes. It follows that M12 is the automorphism group of each of the codes. Arguing similarly as in the proof of Proposition 3.4 we infer that the codes with parameters [220, 110, 20] and [220, 110, 18] are inequivalent. (iii) Here we were not able to establish isomorphism or lack thereof. The proof for items (iv), (v) and (vi) follows similarly to that of item (i), so we omit them. 2 3.3

Extremal self-dual codes invariant under 2-transitive groups

In this section we deal with the question: Given a 2-transitive permutation group G on a set Ω of length n, determine all self-dual extremal codes C of length n on which G acts as a 2-transitive group on the coordinate positions.

480

Rodrigues, Shumba: Almost simple groups and self-dual codes

Due to Mallows-Sloane [25] and Rains [20] a binary self-dual code C of length n and minimum distance d satisfies

n 4 24 + 4, if n ≡ 22 (mod 24) d≤ (1) n 4 24 + 6, if n ≡ 22 (mod 24). A code C is called extremal if equality holds in (1). We recall that the rank of a permutation group G transitive on a set Ω is the number of orbits of Gω , ω a point of Ω, in Ω. Hence, a transitive group G has rank 2 on the set Ω if and only if G is 2-transitive on Ω. The following well-known result about 2-transitive groups can be found in [16]. Lemma 3.6 Every 2-transitive group has a unique minimal normal subgroup (i.e., the socle) which is either elementary abelian or nonabelian simple. Based on this result and assuming that C is an extremal doubly even self-dual code acted on by a 2-transitive group, Malevich and Willems [19] gave the following partial classification. Theorem 3.7 [19, Theorem 1] Let C be an extremal self-dual doubly even code admitting a 2-transitive automorphism group. Then C is either the extended quadratic residue code code of length 8, 24, 32, 48, 80 or 104, or the second Reed-Muller code of length 32, or possibly a code of length 1024 invariant under the group E  SL(2, 25 ), where E is an elementary abelian group of order 210 . Note that Theorem 3.7 conjectures the existence of a putative extremal self-dual doubly even code of length 1024 invariant under the group E  SL(2, 25 ), where E is an elementary abelian group of order 1024. This led Chigira, Harada and Kitazume [6] to complete the classification of extremal self-dual doubly even codes admitting a 2-transitive automorphism group, and thus proving Lemma 3.8 [6, Lemma 1] There is no self-dual code of length 22lt invariant under 2t a group G = E  H where E = (2l ) and H = SL(2t, 2l ). For l and t positive integers, the subgroup E of the group G in Lemma 3.8 is of H. Observe that E acts regularly on itself described as the natural module F2t 2l and H acts on E as the stabilizer of the unit element of E, which is regarded as the zero vector of F2t . Then G acts 2-transitively on E. 2l Observe that the paper [19] (see also [18]) does not give a complete answer to the problem of enumeration and classification of self-dual codes admitting a 2-transitive group as a permutation group of automorphisms. This question is dealt with in [27].

4

Codes invariant under rank 3 almost simple groups

Observe that Theorem 3.7 and Lemma 3.8 answer the question for when the group G has rank 2. Motivated by this, it seems natural to ask: which self-dual codes

Rodrigues, Shumba: Almost simple groups and self-dual codes

481

have rank 3 permutation groups acting on them? Consequentially, which of these self-dual codes (singly even or doubly even), if any exist, are extremal? In this section we deal with these questions restricting the problem to the field κ = F2 . We refer the reader to [26] for a complete treatment of the problem over algebraically closed fields κ of characteristic 2. Recall that a transitive group G has rank 3 if and only if for every point ω in Ω, Gω has two orbits besides {ω}. Rank 3 groups can be either primitive or imprimitive. The following result proved in [26] concerns the existence of self-dual codes that admit a primitive rank 3 group of almost simple type as an automorphism group. Theorem 4.1 Let G be a finite primitive permutation group of almost simple type in its rank 3 action on a set Ω of even degree n. Let κ = F2 and κΩ the κGpermutation module of G on Ω. Further, let C ⊆ κΩ be a self-dual code of length n. Then G is an automorphism group of C if and only if the following hold: (i) q = pt is an odd power of a prime p and G is isomorphic to PSp(2m, q), m ≥ 2 2m −1 q 2m −1 with q ≡ −1 (mod 8) and C is a code with parameters: [ q q−1 , 2(q−1) , d] and d ≥ q + 1. Moreover, C is a doubly even code. (ii) G ∼ = HJ and C is equivalent to one of [100, 50, 10] or [100, 50, 16] code. (iii) G ∼ = HJ:2 and C is equivalent to the unique [100, 50, 10] code. (iv) G ∼ = Ru and C is equivalent to one of three inequivalent [4060, 2030, d] codes with d ≥ 1756.

Remark 4.2 (i) The result of part (i) of Theorem 4.1 for G = PSp(2m, q), with m ≥ 2 follows from the submodule structure of the permutation module κL (L is the set of singular points of PG(l, q)) as well as its complete κGsubmodule lattice were given by Lataille, Sin and Tiep in [17]. Taking X = C(G, L) be the code of the symplectic graph and Y the doubly even code of codimension 1 in X. From [17, Theorem 2.13] we have for m even and m m−1 +1) + 1. Moreover, between X ⊥ and X are m odd dim(X) = q (q −1)(q 2(q−1)

the submodules: X ⊥ , Y ⊥ , U1 , U2 , Y and X, where U1 and U2 are the Weil submodules, and dim(Y ⊥ ) = dim(X ⊥ ) + 1 and dim(Y ) = dim(X) − 1, and q 2m −1 dim(U1 ) = 2(q−1) = dim(U2 ). However, from [17, Remark 2.15] we have that the Weil submodules U1 and U2 exist if and only if q ≡ ±1 (mod 8).

(ii) Observe that parts (ii) and (iii) of Theorem 4.1 appear in Table 7 where the concerned groups are viewed as almost simple sporadic groups. Here these groups are viewed in their primitive rank 3 action on 100 points. The codes were first found by Chigira, Harada and Kitazume in [5]. As illustration of part (iv) of Theorem 4.1 we give the following example. Example 4.3 For G = Ru, let |Ω| = 4060 where Ω is the set of cosets of 2F4 (2) in Ru. Notice that the 2-modular character table of the group Ru is completely

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known (see [23]) and it follows from it that the irreducible 28-dimensional F2 representation is unique. Using the decomposition matrix given in [3] and the ATLAS [7, p. 126] we obtain that the 2-Brauer permutation character of this representation is given as ϕ4060 = 8ϕ1 + 2ϕ28 + 4ϕ376 + 2ϕ1246 . From this we see that there at least two linear combinations of the Brauer characters which give a submodule of dimension 2030, namely ϕ20301 = 4ϕ1 + ϕ281 + 2ϕ3761 + ϕ12461 and ϕ20302 = 4ϕ1 + ϕ282 + 2ϕ3762 + ϕ12462 . By [14, Theorem 2.1] we have that there exists a self-dual code C ⊆ κn with G ≤ Aut(C) if and only if every self-dual simple κG-module U occurs in the κGmodule κn with even multiplicity. Notice that this is indeed the case here. So, helped by computations with Magma [1] we find three submodules of dimension 2030 in the permutation module of degree 4060 of the Rudvalis group over κ = F2 . Example 4.3 gives rise to the following. Proposition 4.4 Up to isomorphism there exist 3 self-dual codes of length 4060 invariant under G = Ru over F2 . Proof That the submodules are indeed self-dual codes follows by [4, Lemma 2.11]. By [21, Lemma 8], let C(G, Ω)⊥ be the code of the 28-dimensional irreducible representation of Ru over κ = F2 . We show that there are three inequivalent self-dual codes between C(G, Ω)⊥ and C(G, Ω), say M1 , M2 and M3 . Since a selfdual code of even length contains the all-one codeword, using [21, Lemma 6] we deduce that these codes Mi have minimum weight at least 1756 as they contain the code spanned by the rows of the adjacency matrix of the graph adjoined by the all-ones vector. The group G acts on the set {M1 , M2 , M3 } of codes Mi . By the Orbit-Stabilizer Theorem, we have [G : GMi ] ≤ 3, using the fact that the set {M1 , M2 , M3 } has three elements. Since G has no subgroups of index ≤ 3 it follows that each Mi , 1 ≤ i ≤ 3 is G-invariant. Now, we determine the the automorphism group Aut(Mi ). Since for each i where 1 ≤ i ≤ 3 we have Mi is Ginvariant and Out(G) = 1 it follows that a group of degree 4060 containing Ru is one of Ru, A4060 or S4060 . However, since each Mi must contain the all-ones codeword then it contains the subcode of dimension 29 invariant under Ru, and by [21, Lemma 5] we exclude the symmetric and alternating groups as an automorphism group of a self-dual code of length 4060. Further, G = Aut(C(G, Ω)), whence G ⊆ Aut(Mi ) ⊆ Aut(C(G, Ω)) = G. Therefore Aut(Mi ) = G for i = 1, 2, 3 and by [4, Lemma 2.3], the three codes are inequivalent. 2 The imprimitive rank 3 groups have been classified in [10] (see [11] for the corrected version) and it follows from this that there are two infinite families and a finite number of individual imprimitive examples of rank 3 groups of almost simple type. In the theorem below, proved in [28] we present results on self-dual codes invariant under imprimitive rank 3 groups of almost simple type, confining ourselves to lengths n ≤ 4095.

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Theorem 4.5 Let G be an imprimitive rank 3 group of almost simple type of degree at most 4095 on a set Ω of even degree n. Let κ = F2 and κΩ the κGpermutation module of G on Ω. Let C ≤ κΩ be a self-dual code of length n. Then G is an automorphism group of C if and only if G is isomorphic to one of the groups: 211 S11 of degree 22, Aut(M12 ) of degree 24, PSL(4, 9) of degree 1640 or PSL(3, 2) of degree 14 and C is a code with parameters: [22, 11, 2]2 , [24, 12, 8]2 , [1640, 820, d]2 , d < 276, or [14, 7, 2]2 . In Theorem 4.6 we enumerate self-dual doubly even codes invariant under rank 3 groups of almost simple type and finally in Theorem 5.1 we give a result on the existence of extremal self-dual codes invariant under almost simple groups of sporadic or rank 3 types. Theorem 4.6 Let C be a self-dual doubly even code admitting a rank 3 automorphism group G of almost simple type. Then C is a code with parameters 2m −1 q 2m −1 [ q q−1 , 2(q−1) , d] with q ≡ −1 (mod 8) and ≥ q + 1, [1640, 820, d], d < 276 or the extended binary Golay code and G is isomorphic to PSp(2m, q), m ≥ 2 and q ≡ −1 (mod 8), PSL(4, 9), and M12 :2, respectively. We mention here that the self-dual binary codes from the classes of rank 3 permutation groups of grid and affine types, respectively are the subject of forthcoming papers.

5

Extremal codes

In this section we classify extremal binary self-dual codes admitting an almost simple group G of sporadic or rank 3 type as a permutation automorphism group. Recall that a binary self-dual code C is called doubly even, also known as type II, if all its weights are divisible by 4. By a result of Gleason [13], the length n of a self-dual doubly even code is a multiple of 8. If C is an extremal doubly even n code then d ≤ 4 24 + 4 (see [20]) and n ≤ 3928, by a result of Zhang [29]. The bound for the length of singly-even extremal self-dual codes is still open. However, the existence of extremal doubly even codes is known only for small values of n; the largest being 136. Thus, there is a large gap between the bound on the length of doubly even extremal codes and what we can really construct. From the examples of self-dual codes (singly-even and doubly even, respectively) constructed in this paper (see Theorems 3.3, 4.1, 4.5 and 4.6) we verify which ones satisfy (1). Recall that self-dual doubly even codes are bounded on the distance and on the length. A simple check of the minimum distance shows that there are possibilities of existence of self-dual extremal codes in the following instances: Aut(M11 ) of degree 24, PSp(4, 7) of degree 400 (see Theorem 4.1(i) with q = 7), and PSL(4, 9) of degree 1640. Thus, the classification of extremal codes invariant under an almost simple group G of sporadic or rank 3 type as a permutation automorphism group is reduced to determining whether the codes of lengths 24, 400 and 1640, respectively are extremal. In the following we give a result on extremal codes.

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Theorem 5.1 Let C be an extremal self-dual code of length n ≤ 4095 admitting an almost simple group G of sporadic or rank 3 type as a permutation automorphism group. Then C is either the extended binary Golay code or C is a [44, 22, 8]2 -subcode of C(M22 :2, 44) = [44, 33, 2] and G = M12 :2, M24 or G = M22 :2. In particular, up to isomorphism, the C(M22 : 2, 44)-subcode C = [44, 22, 8]2 is a Type I code. Proof By Theorem 3.3 and Theorem 4.6 we have that equality holds in (1) for n = 24 and n = 44. For n = 24 we have C is the extended binary Golay [24, 12, 8] code and C is invariant under G = M12 :2 and G = M24 respectively. Now, it follows from [19, Theorem 1] that C is an extremal quadratic residue code. For n = 44 it follows from Table 4, that C = [44, 22, 8] and G = M22 :2. 2 Acknowledgements This work is based on the research supported by the National Research Foundation of South Africa (Grant Numbers 95725 and 106071). References [1] W. Bosma, J. Cannon, C. Playoust. The Magma algebra system I: The user language. J. Symbolic Comput., 24 (1997), 235–265. [2] W. Bosma and J. Cannon. Handbook of Magma Functions. Department of Mathematics, University of Sydney, 1994. http://www.maths.usyd.edu.au:8000/u/magma/. [3] Thomas Breuer. Decomposition Matrices. Available at: The Modular Atlas homepage. http://www.math.rwth-aachen.de/∼MOC/decomposition 1999. [4] N. Chigira, M. Harada and M. Kitazume. Permutation groups and binary selforthogonal codes. J. Algebra, 309 (2007), 610–621. [5] N. Chigira, M. Harada and M. Kitazume. Some self-dual codes invariant under the Hall-Janko group. J. Algebra, 316 (2007), 578–590. [6] N. Chigira, M. Harada and M. Kitazume. On the classification of extremal doubly even self-dual codes with 2-transitive automorphism groups. Des. Codes Cryptogr., 73 (2014), 33–35. [7] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson. An Atlas of Finite Groups. Oxford: Oxford University Press, 1985. [8] T. Connor and D. Leemans. An atlas of subgroup lattices of finite almost simple groups. Ars Math. Contemp., 8 (2015), 259–266. [9] T. Connor and D. Leemans. The Subgroup Lattice of M11 . http://homepages.ulb.ac.be/∼dleemans/atlaslat/m11.pdf. [10] A. Devillers, M. Giudici, C. H. Li, G. Pearce and C. E. Praeger. On imprimitive rank 3 permutation groups. J. London Math. Soc., 84 (2011), 649-–669. [11] A. Devillers, M. Giudici, C. H. Li, G. Pearce and C. E. Praeger. Correction to “On imprimitive rank 3 permutation groups,” J. London Math. Soc. 84 (2011), 649-–669. J. London Math. Soc., (2) 85 (2012) 592. [12] GAP. Groups, Algorithms and Programming, Version 4. The GAP Group, Lehrstuhl D f¨ ur Mathematik, RWTH Aachen, Germany and School of Computer Sciences, University of St Andrews, Scotland. http://www.gap-system.org/. [13] A.M. Gleason. Weight polynomials of self-dual codes and the MacWilliams identities. In: Actes du Congr´es International des Math´ematiciens (Nice, 1970), Tome 3, pp. 211–215. Gauthier-Villars, Paris (1971). [14] A. G¨ unther and G. Nebe. Automorphisms of doubly even self-dual codes. Bull. London Math. Soc., 41 (2009), 769–778.

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[15] W. Cary Huffman. On the classification and enumeration of self-dual codes. Finite Fields and Their Applications, 11 (2005), 451–490. [16] B. Huppert. Endliche Gruppen I. Berlin, Heidelberg: Springer Verlag, 1967. [17] J. M. Lataille, Peter Sin and Pham Huu Tiep. The modulo 2 structure of rank 3 permutation modules for odd characteristic symplectic groups. J. Algebra, 268 (2003), 463–483. [18] A. Malevich. Extremal self-dual codes. Ph.D. Thesis, Otto-von-Guericke-Universit¨at Magdeburg, 2012. [19] A. Malevich and W. Willems. On the classification of the extremal self-dual codes over small fields with 2-transitive automorphism groups. Des. Codes Cryptogr. 70 (2014), 69–76. [20] C. L. Mallows and N.J.A. Sloane. An upper bound for self-dual codes. Inf. Control, 22 (1973), 188–200. [21] J. Moori and B. G. Rodrigues. Some designs and binary codes preserved by the simple group Ru of Rudvalis. J. Algebra, 372 (2012), 702–710. [22] S. Mukwembi, B. G. Rodrigues and T. M. Shumba. On self-dual binary codes invariant under permutation groups. Submitted. [23] R. A. Parker and R. A. Wilson. The 2-modular character table of the Rudvalis group, unpublished, 1998. [24] E. M. Rains. Shadow bounds for self-dual-codes. IEEE Trans. Inf. Theory, 44 (1998), 134–139. [25] E.M. Rains and N.J.A. Sloane. Self-dual codes. In V. S. Pless and W. C. Huffman, editors, Handbook of Coding Theory, pages 177–294. Amsterdam: Elsevier, 1998. Volume I and II. [26] B. G. Rodrigues. A classification of binary self-dual codes with a primitive rank 3 automorphism group of almost simple type. Submitted. [27] B. G. Rodrigues. On self-dual codes invariant under 2-transitive groups. In preparation. [28] B. G. Rodrigues. A classification of binary self-dual codes admitting an imprimitive rank 3 automorphism group of almost simple type. Submitted. [29] S. Zhang. On the nonexistence of extremal self-dual codes. Discrete Appl. Math., 91 (1999), 277–286.

TEST ELEMENTS: FROM PRO-p TO DISCRETE GROUPS ILIR SNOPCE∗ and SLOBODAN TANUSHEVSKI† ∗

Instituto de Matem´ atica, Universidade Federal do Rio de Janeiro, 21941 - 909, Rio de Janeiro, RJ, Brasil Email: [email protected]

†

Instituto de Matem´ atica e Estat´ıstica, Universidade Federal Fluminense, 24210 - 201, Niter´ oi, RJ, Brasil Email: [email protected]

Abstract We survey some recent results on test elements of pro-p groups and discrete groups. An element g of a group G is a test element of G if every endomorphism of G that fixes g is an automorphism. Our main focus is on pro-p techniques in the study of test elements of discrete groups. In particular, we discuss the distribution of test elements in free groups and surface groups.

1

Introduction

An element g of a group G is called a test element of G if every endomorphism of G that fixes g is an automorphism. Equivalently, g ∈ G is a test element if for every ϕ ∈ End(G), the existence of α ∈ Aut(G) such that ϕ(g) = α(g) implies ϕ ∈ Aut(G). The notion of a test element was introduced by Shpilrain in [36]. For n ≥ 1, let Fn denote the free group on {x1 , . . . , xn } and denote the orientable surface group of genus n by G(n) = x1 , . . . , x2n | [x1 , x2 ] . . . [x2n−1 , x2n ] . • The commutator [x1 , x2 ] is a test element of F2 (Nielsen, [28]). • [x1 , x2 ][x3 , x4 ] · · · [x2m−1 , x2m ] is a test element of F2m (Zieschang, [47]). • Every higher commutator of weight n (with arbitrary disposition of commutator brackets) involving all n letters x1 , ..., xn is a test element of Fn (Rips, [33]). • xk11 xk22 · · · xknn is a test element of Fn if and only if ki = 0 for all 1 ≤ i ≤ n and gcd(k1 , . . . , kn ) = 1 (Turner, [43]). • xk1 xk2 . . . xk2n is a test element of G(n) for every k ≥ 2 and n ≥ 2 (O’Neill and Turner, [30]). For more examples of test elements (in different classes of groups), as well as for a discussion of other related concepts, we refer the reader to the book [26]. A retract of a group G is a subgroup R ≤ G for which there exists a homomorphism r : G → R, called a retraction, such that r(g) = g for all g ∈ R. Clearly, if an element g ∈ G belongs to a proper retract of G then it is not a test element. Turner proved that the converse is also true for free groups of finite rank.

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Theorem 1.1 (Turner [43]) Let F be a free group of finite rank and let w ∈ F . Then w is a test element of F if and only if w does not belong to a proper retract of F . A group G is called a Turner group if the above theorem holds for G, that is, an element g ∈ G is a test element of G if and only if it is not contained in any proper retract of G. O’Neill and Turner proved that torsion free stably hyperbolic groups and finitely generated Fuchsian groups are also Turner groups [30]. More recently, Groves [14] proved that all torsion free hyperbolic groups are in fact Turner groups. The fundamental group of the Klein bottle a, b | aba−1 = b−1 is an example of a group which is not a Turner group (see [43]). However, all other surface groups are Turner groups. In this article, we survey some recent results on test elements of profinite groups and discrete groups based primarily on [15], [38], [39], and [40]. All finitely generated profinite groups are Turner groups. The observation that the only retracts of a free pro-p group are the free factors leads to a tame theory of test elements of free pro-p groups of finite rank. Moreover, to a great extent the study of test elements of Demushkin groups can be reduced to that of free pro-p groups. Every element of a residually finite-p Turner group G that is a test element of the pro-p completion of the group is also a test element of G. This fact opens up a way for pro-p techniques to be used in the study of test elements of discrete groups. Indeed, the results on test elements of free pro-p groups and Demushkin groups have strong implications for the distribution of test elements of free (discrete) groups and surface groups.

2

Test elements of profinite groups

A profinite group is a compact totally disconnected topological group. In particular, all finite groups are profinite (with respect to the discrete topology). Furthermore, it can be shown that the category of profinite groups coincides with the category of topological groups that are inverse limits of inverse systems of finite groups. The Galois group of an infinite Galois field extension equipped with the Krull topology is a profinite group. Moreover, every profinite group can be realized as the Galois group of some Galois extension ([21]). Profinite groups also appear in algebraic geometry as ´etale fundamental groups of schemes. Henceforth, we adopt the following conventions. A subgroup H of a profinite group G is tacitly taken to be closed. If G is a profinite group and X ⊆ G, then X is the closed subgroup of G topologically generated by X. Homomorphisms between profinite groups are always assumed to be continuous. However, it follows from a theorem of Nikolov and Segal that every homomorphism from a (topologically) finitely generated profinite group to any profinite group is continuous ([29]). Throughout, p denotes a fixed prime. A profinite group in which every open subgroup has index a power of p is called a pro-p group. The additive group of the ring of p-adic integers Zp is a well-known example of a pro-p group. A succinct introduction to profinite and pro-p groups is given in [9, Chapter 1]. For a more

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comprehensive exposition, we refer the reader to [32] and [44]. The Frattini subgroup Φ(G) of a non-trivial profinite group G is defined to be the intersection of all maximal proper open subgroups of G; and Φ(G) = G if G is the trivial group. A subset X of G topologically generates G if and only if G/Φ(G) = XΦ(G)/Φ(G) ([9, Proposition 1.9]). If G is a finitely generated prop group, then Φ(G) = Gp [G, G] ([9, Corollary 1.20]), and thus G/Φ(G) is a finite dimensional vector space over Fp , the finite field with p elements; it follows that the minimal number of generators of G , denoted by d(G), is exactly dimFp G/Φ(G). A point of departure for the study of test elements of profinite groups is provided by the following theorem. Theorem 2.1 ([38]) Every finitely generated profinite group is a Turner group. It is not difficult to prove the above theorem for finite groups. Then a standard inverse limit argument can be used to extend the result to all finitely generated profinite groups. However, in [38], we give a slightly more involved proof which in return reveals more, namely, that for each endomorphism ϕ: G → G of a finitely n generated profinite group G, the stable image ϕ∞ (G) = ∞ n=1 ϕ (G) of ϕ is a retract of G. How many test elements can a profinite group have? To begin with, there are profinite groups that do not have any test elements. Examples of such groups are provided by the non-cyclic free abelian pro-p groups of finite rank. To see this, let G = Znp be the free abelian pro-p group of rank n ≥ 2, and let g be a non-trivial element of G. If g ∈ / Φ(G), then g is clearly a proper retract of G, so g is not a test element. Now suppose that g ∈ Φ(G) = Gp . Then there is a positive integer k k k+1 such that g ∈ Gp \ Gp . By unique extraction of roots, there exists h ∈ G such k that g = hp . Obviously, h ∈ G \ Gp , and thus h is a proper retract of G. Hence, once again, g is not a test element of G. On the other hand, the finitely generated just infinite profinite groups tend to have many test elements. A profinite group is termed just infinite if it is infinite and all of its proper quotients are finite. For d ≥ 2, the groups SLd (Zp ) and SLd (Fp [[t]]), where Fp [[t]] is the pro-p ring of formal power series over a finite field with p elements, are examples of just infinite profinite groups. Note that these groups are virtually pro-p. Indeed, many of the well-known just infinite pro-p groups are groups of Lie type defined over Zp (i.e., p-adic analytic groups) or over Fp [[t]]. In particular, the first congruence subgroups SL1d (Zp ) = ker(SLd (Zp ) → SLd (Zp /pZp )) and SL1d (Fp [[t]]) = ker(SLd (Fp [[t]]) → SLd (Fp [[t]]/tFp [[t]])) are just infinite pro-p groups. The Nottingham group N (Fp ), which can be described as the group of automorphisms of Fp [[t]] that act trivially on tFp [[t]]/t2 Fp [[t]], is an example of a just infinite pro-p group that is not linear over any profinite ring. The most remarkable property of N (Fp ) is that it contains a copy of every countably based pro-p group. A careful treatment of just infinite profinite groups is given in [45]. For a thorough discussion of p-adic analytic just infinite pro-p groups see [17]. Corollary 2.2 In a finitely generated just infinite profinite group every element of infinite order is a test element. In particular, in a torsion free finitely generated

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just infinite profinite group every non-trivial element is a test element. Proof Let G be a finitely generated just infinite profinite group, and let g ∈ G be an element of infinite order. Suppose that g is not a test element of G. By Theorem 2.1, there is a proper retract R of G containing g. However, every retraction r : G → R has a non-trivial kernel, which implies that R is an infinite proper quotient of G, a contradiction with the fact that G is just infinite.  Note that if G is a just infinite pro-p group, then it is finitely generated since G/Φ(G) is finite. Thus the above result holds for all just infinite pro-p groups. Before we consider test elements in other families of profinite groups, we give some indication of the importance of test elements in the study of pro-p groups. 2.1

Endomorphisms of pro-p groups preserving an automorphic orbit

Given a group G, we denote by O(g) the automorphic orbit of an element g ∈ G, that is, O(g) = {α(g) ∈ G | α ∈ Aut(G)}. Let ϕ be an endomorphism of G and g ∈ G be a non-trivial element. An obvious necessary condition for ϕ to be an automorphism is that ϕ(O(g)) ⊆ O(g). The following theorem of Lee, conjectured by Shpilrain in [36], states that in the case when G is a free group of finite rank this is also a sufficient condition. Theorem 2.3 (Lee [20]) Let F be a free group of finite rank, and let ϕ be an endomorphism of F . If ϕ(O(w)) ⊆ O(w) for some w ∈ F, w = 1, then ϕ is an automorphism. Remark 2.4 Theorem 2.3 was proved for the free group of rank two by Shpilrain [37] and Ivanov [11]. Building upon the work of Ivanov, Lee first proved in [19] that the theorem is valid when w is a primitive element of F , that is, when ϕ is a primitivity preserving endomorphism. Finally, in [20], Lee showed that ϕ(O(w)) ⊆ O(w) for any w = 1 implies that ϕ is primitivity preserving. Lee’s proof uses combinatorial techniques (Nielsen reduced sequences, Whitehead automorphisms, Whitehead graphs, etc.) none of which have counterparts in the category of pro-p groups. However, an argument combining Theorem 2.1 with further results on retracts of pro-p groups and Frattini arguments, yields a stronger result in the category of pro-p groups. Theorem 2.5 ([38]) Let G be a finitely generated pro-p group, and suppose that Aut(G) acts transitively on G/Φ(G). If ϕ is an endomorphism of G such that ϕ(O(g)) ⊆ O(g) for some g = 1, then ϕ is an automorphism. Proof Suppose that ϕ is an endomorphism of G satisfying ϕ(Orb(g)) ⊆ Orb(g) for some g = 1. We will show that ϕ maps primitive elements (i.e., elements belonging to G \ Φ(G)) to primitive elements. The theorem then follows from

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the easy observation that every endomorphism that maps primitive elements to primitive elements is an automorphism (see [38, Lemma 4.3]). There exists a retract H of G minimal among retracts of G that contain g (see [38, Lemma 4.1]). Consequently, g is not contained in any proper retract of H. Being a retract of a finitely generated group, H is finitely generated. Hence, by Theorem 2.1, g is a test element of H. Let r : G → H be a retraction. Since ϕ(Orb(g)) ⊆ Orb(g), there exists α ∈ Aut(G) such that ϕ(g) = α(g). Consider the homomorphism β : H → H defined by β := r ◦ (α−1 ◦ ϕ)|H . We have β(g) = r(α−1 (ϕ(g))) = r(α−1 (α(g))) = r(g) = g. As g is a test element of H, it follows that β is an automorphism of H. Let x ∈ H be a primitive element of H. Using that H is a retract of G, it is easy to show that Φ(H) = H ∩ Φ(G). Hence, x ∈ / Φ(G) and thus x is a primitive element of G. Furthermore, β(x) is a primitive element of H and (α−1 ◦ ϕ)(x) is a primitive element of G. Indeed, (α−1 ◦ ϕ)(x) ∈ Φ(G) implies β(x) = r((α−1 ◦ ϕ)(x)) ∈ Φ(H), a contradiction with the primitivity of β(x). As α is an automorphism of G, it follows that ϕ(x) = α((α−1 ◦ ϕ)(x)) is a primitive element of G. Now let y ∈ G be an arbitrary primitive element of G. Since Aut(G) acts transitively on G/Φ(G), there is γ ∈ Aut(G) such that γ(x)y −1 ∈ Φ(G). Observe that γ(H) is a retract of G and that it is minimal among retracts of G that contain γ(g) . Moreover, γ(x) is a primitive element of γ(H) . Hence, we can repeat the argument from the previous paragraph with g replaced by γ(g) (Orb(γ(g)) = Orb(g)) and conclude that ϕ(γ(x)) is a primitive element of G. From ϕ(γ(x))ϕ(y)−1 = ϕ(γ(x)y −1 ) ∈ Φ(G), we get that ϕ(y) is a primitive element of G. Hence, ϕ maps primitive elements to primitive elements.  In particular, the above theorem holds for free pro-p groups of finite rank (free pro-p groups are discussed in more details in the next subsection). Corollary 2.6 Let F be a relatively free pro-p group of finite rank. If ϕ is an endomorphism of F such that ϕ(O(w)) ⊆ O(w) for some w = 1, then ϕ is an automorphism. In particular, this statement is true for free pro-p groups of finite rank. In [38], an example is given of an endomorphism ϕ of a finitely generated pro-p group G that is not an automorphism and for which there exists an element g ∈ G, g = 1, such that ϕ(O(g)) ⊆ O(g). It would be nice to know exactly for which pro-p groups the conclusion of Theorem 2.5 holds. 2.2

Free pro-p groups

, p of a discrete group G is defined to be the inverse limit of The pro-p completion G the groups G/N where N runs through the normal subgroups of G of index a finite , p induced by the projections power of p. There is a homomorphism ıp : G → G

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G → G/N . If G is residually finite-p, then ıp is an embedding; in this case, we identify ıp (G) with G. The pro-p completion of a free discrete group F with finite basis X is a free pro-p group (a free object in the category of pro-p groups) with basis X (since free discrete groups are residually finite-p, X is naturally identified with a subset of F,p ). Thus Zp , the pro-p completion of Z, is the free pro-p group of rank one. Clearly, every non-trivial element of Zp is a test element. On the other hand, the test elements of a free pro-p group of rank at least two are all contained in the Frattini subgroup. In sharp contrast with free discrete groups, which have many retracts that are not free factors (in a sense that is rendered more precise in Section 4), every retract of a free pro-p group is a free factor. Proposition 2.7 The only retracts of a free pro-p group are the free factors. Hence, the test elements of a free pro-p group F of finite rank are exactly the elements not contained in any proper free factor of F . Proof There is a simple criterion to decide if a subgroup H of a free pro-p group F is a free factor: H is a free factor of F if and only if Φ(F ) ∩ H = Φ(H). Let r : F → R be a retraction. Then Φ(R) ⊆ R ∩ Φ(F ), and using that r is a retraction, we get R ∩ Φ(F ) = r(R ∩ Φ(F )) ⊆ r(R) ∩ r(Φ(F )) = R ∩ Φ(R) = Φ(R). Hence, R ∩ Φ(F ) = Φ(R) and R is a free factor of F .



Let F be a free pro-p group with basis {x1 , . . . , xn }. The following elements of F are test elements: • any higher commutator of weight n involving all n letters x1 , . . . , xn ; •

β

2k−1 1 2 2k xα1 1 . . . xαmm [xβm+1 , xβm+2 ] . . . [xm+2k−1 , xβm+2k ]

where m, k ≥ 0, n = m + 2k, αi ∈ pZp \ {0} for 1 ≤ i ≤ m, and βi ∈ Zp \ {0} for 1 ≤ i ≤ 2k. More concrete examples of test elements of free pro-p groups, as well as general techniques for constructing them, can be found in [38]. The following theorem gives some evidence of the ubiquity of test elements in free pro-p groups. Theorem 2.8 ([38]) Let F be a non-abelian free pro-p group of finite rank. Then the set of test elements of F is a dense subset of Φ(F ). The main step in the proof of Theorem 2.8 is the following lemma, which, as we shall see later, has some other important implications. Lemma 2.9 ([38]) Let F be a pro-p group with basis {x1 , . . . , xn }, and let w ∈ Φ(F ). Then there exists a subset {xi1 , . . . , xim } ⊆ {x1 , . . . , xn } such that for every test element u of H = xi1 , . . . , xim contained in Φ(H), w · u is a test element of F .

492 2.3

Snopce, Tanushevski: Test elements: from pro-p to discrete groups Demushkin groups

Let n ≥ 1 be an integer. A pro-p group G is called a Poincar´e group of dimension n if it satisfies the following conditions: (i) dimFp H i (G, Fp ) < ∞ for all i, (ii) dimFp H n (G, Fp ) = 1, and

(iii) the cup-product H i (G, Fp ) × H n−i (G, Fp ) → H n (G, Fp ) ∼ = Fp is a nondegenerate bilinear form for all 0 ≤ i ≤ n. A Poincar´e group of dimension 2 is called a Demushkin group (see [35]). Demushkin groups play an important role in algebraic number theory. For instance, if k is a p-adic number field containing a primitive p-th root of unity and k(p) is the maximal p-extension of k, then Gal(k(p)/k) is a Demushkin group (see [27]). Demushkin groups have been classified completely by Demushkin, Serre, and Labute ([7], [8], [34], and [18]). The pro-p completion of an orientable surface group is a Demushkin group. Moreover, Demushkin groups have many properties reminiscent of surface groups. For example, every finite index subgroup H of a Demushkin group G is a Demushkin group with d(H) = 2 + [G : H](d(G) − 2), and every subgroup of infinite index is free pro-p (cf. [35] or [27]). Sonn [41] proved that if G is a Demushkin group with minimal generating set of size n and F is a free homomorphic image of G, then F is of rank at most n2 . It follows easily that every retract of a Demushkin group is free pro-p of rank at most n2 . The following theorem shows that test elements of free pro-p groups can be used to construct test elements in Demushkin groups. Theorem 2.10 ([38]) Let G be a Demushkin group with d(G) = n > 2 and {x1 , x2 , !. . . , xn } be a minimal generating set of G. Fix a positive integer k such that n2 + 1 ≤! k ≤ n. For each 1 ≤ i ≤ k, let αi ∈ Zp \ {0} and suppose that at least k − n2 + 1 of the elements α1 , . . . , αk belong to pZp \ {0}. Moreover, if p = 2 and n ≤ 4, suppose that k < n. If w(y1 , y2 , . . . , yk ) is a test element of the free pro-p group F with basis {y1 , y2 , . . . , yk }, then w(xα1 1 , xα2 2 , . . . , xαk k ) is a test element of G.

3

From pro-p groups to discrete groups

, p for the Let G be a finitely generated discrete group. As before, we write G pro-p completion of G. Recall that if G is residually finite-p, then G is naturally , p . Nonetheless, in general, one would not identified with a dense subgroup of G expect to find a strong connection between the test elements of G and those of its pro-p completion; however, when G is a Turner group, we have the following result. Proposition 3.1 ([38]) Let p be a prime, and let G be a finitely generated residu, p , then g is a test element ally finite-p Turner group. If g ∈ G is a test element of G of G. , p , and let R be a proper retract of G. Fix Proof Let g ∈ G be a test element of G a retraction r : G → R, and let j : R → G denote the inclusion homomorphism.

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After applying the pro-p completion functor, we obtain the following commutative diagram of pro-p groups: ,p R

jp

id

,p /G  

rp

,p R

jp

,p /R⊆G

,p ) = R ([32, Theorem 3.2.4] ). It follows that jp is an injection. Moreover, jp (R , p → R is a For each jp (x) ∈ R, we have (jp ◦ rp )(jp (x)) = jp (x). Hence, jp ◦ rp : G retraction. Since R is a proper retract of G, r is not injective. Consequently, rp is , p . Since g is a test element of G ,p , also not injective and R is a proper retract of G it follows that g ∈ / R. Therefore, g ∈ / R, and it follows from Theorem 2.1 that g is a test element of G.  Non-orientable surface groups of genus n ≥ 4 and all orientable surface groups are residually free, and therefore residually finite-p (for every prime p) since free groups are residually finite-p (see [3] and [2]). Furthermore, the non-orientable surface group of genus 3 is also residually finite-p, by [22, Lemma 8.9]. Hence, free groups, non-orientable surface groups of genus n ≥ 3, and all orientable surface groups satisfy the hypothesis of Proposition 3.1. As was already mentioned above, the pro-p completions of free groups of finite rank and orientable surface groups are free pro-p groups and Demushkin groups, respectively. Moreover, it is not difficult to see that for every prime p = 2 the pro-p completion of a non-orientable surface group is also a free pro-p group. Therefore, we can apply the results of the previous section to obtain many concrete examples of test elements in these classes of groups. Further application concerning the distribution of test elements will be given in Section 4. 3.1

From free pro-p to free discrete

Let F be a free group of finite rank. Given a test element w of F , does there always exist a prime p such that w is also a test element of F,p ? The answer is positive in the case when F has rank two. Proposition 3.2 ([38]) Let F be a free group of rank two, and let w ∈ F . Then w is a test element of F if and only if there exists a prime p such that w is a test element of F,p . In contrast, every free group of rank n ≥ 3 has test elements that are not test elements of any pro-p completion of the group. In order to give concrete examples, we state a theorem of Fine, Rosenberger, Spellman, and Stille. Let F be a free group with basis {x1 , . . . , xn }. For each 1 ≤ i ≤ n, let σxi : F → Z be the homomorphism defined by σxi (xj ) = δi,j , where δi,j is the Kronecker delta. Thus for w ∈ F , σxi (w) is the sum of the exponents of all occurrences of xi in w.

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Theorem 3.3 (Fine et al. [10]) Let X = X1  X2  . . .  Xr be a finite set partitioned into r ≥ 2 non-empty subsets. Let F be the free group on X, and let ui ∈ Xi ≤ F, 1 ≤ i ≤ r, be such that gcd(

r 

{σx (ui ) | x ∈ Xi }) = 1.

i=1

Then u1 u2 . . . ur is a test element of F if and only if ui is a test element of Xi for all 1 ≤ i ≤ r. Let X1 = {x1 , x2 , . . . , xm }, X2 = {xm+1 , xm+2 , . . . , xm+n } (n, m ≥ 2), and let F be the free group on X = X1 ∪ X2 . Then (x21 x22 . . . x2m )5 and (x3m+1 . . . x3m+n )10 are test elements of X1 and X2 , respectively. It follows from Theorem 3.3 that w = (x21 x22 . . . x2m )5 (x3m+1 . . . x3m+n )10 is a test element of F . However, w is not a test element of F,p for any prime p. Indeed, if p = 2, then u = x21 x22 . . . x2m is a primitive element of F,p and u, X2 ≤ F,p is a proper free factor of F,p containing w; similarly, v = x3m+1 . . . x3m+n is a primitive element of F,2 and w is contained in the proper free factor v, X1 of F,2 . Although, a complete classification of test elements of free groups seems, at least at the moment, to be out of reach, we believe that the line of investigation suggested by Theorem 3.3 is promising. Problem 3.4 Let X = X1  X2  . . .  Xr be a finite set partitioned into r ≥ 2 non-empty subsets, and let F be the free group on X. For i = 1, 2, . . . , r, let ui ∈ Xi ≤ F be such that gcd(

r 

{σx (ui ) | x ∈ Xi }) = 1.

i=1

Let w(z1 , . . . , zr ) be a group word in the variables z1 , . . . , zr . Find sufficient and necessary conditions on ui (1 ≤ i ≤ r) and w for w(u1 , . . . , ur ) to be a test element of F . Problem 3.4 has been solved in the case when r = 2. Theorem 3.5 ([40]) Let X = X1  X2 be a finite set partitioned into two disjoint non-empty subsets, and let F be the free group on X. Let u1 ∈ X1 and u2 ∈ X2 be such that d = gcd({σx (u1 ) | x ∈ X1 } ∪ {σx (u2 ) | x ∈ X2 }) = 1. Let w(z1 , z2 ) be a group word in the variables z1 , z2 . Then w(u1 , u2 ) ∈ F is a test element of F if and only if w is neither a power of a conjugate of z1 nor z2 , and u1 and u2 are test elements of X1 and X2 , respectively. The general case, r ≥ 3, of Problem 3.4 is widely open (however, some partial results were given in [40]).

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4

495

Distribution of test elements in free and surface groups

Let G be a finitely generated group with a finite generating set X. Given g ∈ G, we denote by |g|X the word norm of g with respect to X, that is, |g|X is the smallest integer k ≥ 0 for which there exist x1 , . . . , xk ∈ X ±1 such that g = x1 . . . xk . The word metric on G with respect to X can be defined by dX (g, h) = |g −1 h|X for g, h ∈ G. A subset S of G is called a C-net (0 ≤ C < ∞) with respect to X if dX (g, S) = inf{dX (g, s) | s ∈ S} ≤ C for all g ∈ G. In other words, S is a C-net if the balls of radius C centered at the elements of S cover the entire group. Theorem 4.1 ([39]) The set of test elements of a free group F of rank n ≥ 2 is a (3n − 2)-net with respect to every finite generating set of F . The main ingredients in the proof of Theorem 4.1 are Proposition 3.1 and Lemma 2.9. Since for p = 2, the pro-p completion of a non-orientable surface group is free pro-p, the same techniques lead to the following result. Theorem 4.2 ([39]) The set of test elements of a non-orientable surface group G of genus n ≥ 3 is a (5n − 5)-net with respect to every finite generating set of G. The following lemma is an analogue of Lemma 2.9 for Demushkin groups. Lemma 4.3 ([39]) Let G be a Demushkin pro-p group with p ≥ 5 and d(G) > 2. Let {a1 , . . . , a2n } be a minimal generating set of G. If w ∈ Φ2 (G) = Φ(Φ(G)), then there exists a subset {ai1 , . . . , aim } ⊆ {a1 , . . . , an+1 } such that for every test element u(x1 , . . . , xm ) of the free pro-p group F,p with basis {x1 . . . . , xm } contained in Φ(F,p ), w · u(api1 , . . . , apim ) is a test element of G. Lemma 4.3 plays a key role in the proof of the following theorem. Theorem 4.4 ([39]) The set of test elements of an orientable surface group G of genus n ≥ 2 is a (161n + 8 · 25n (n − 1)(16n + 1) + 33)-net with respect to every finite generating set of G. Let G be a finitely generated group with a finite generating set X. We denote by BX (r) the ball of radius r ≥ 0 centered at the identity in the metric space (G, dX ). Given S ⊆ G, the asymptotic density of S in G with respect to X is defined as ρX (S) = lim sup k→∞

|S ∩ BX (k)| . |BX (k)|

If the actual limit exists, we refer to it as the strict asymptotic density of S in G with respect to X, and we write ρX (S) instead of ρX (S). A subset S of G is generic in G (with respect to X) if ρX (S) = 1, and it is negligible if ρX (S) = 0. Clearly, a subset is generic if and only if its complement is negligible.

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Let F be a free group of rank n ≥ 2. The following are some examples of negligible subsets of F : the set of all proper powers [1]; the set of primitive elements [6], [5]; any finite union of automorphic orbits [13]. For other examples of negligible subsets of F see [46] and [16]. On the other hand, the set of elements of F with cyclic stabilizers in Aut(F ) [16] and the set of words with nontrivial images in the abelianization of F [46] are examples of generic subsets of F . Most of the subsets of free groups for which the asymptotic density has been studied and which could be defined by a natural algebraic property are either generic or negligible. Therefore, it is interesting to find examples of some natural subsets that have intermediate density (different from 0 and 1). Such an example was given in [15], where it was proved by Kapovich, Rivin, Schupp and Shpilrain that the set of test elements of a free group of rank two has intermediate density. Theorem 4.5 (Kapovich et al. [15]) Let T be the set of test elements of a free group F with basis {x1 , x2 }. Then T has intermediate density. More precisely, 4 6 8 (1 − 2 ) ≤ ρ{x1 ,x2 } (T ) ≤ 1 − 2 . 9 π 3π In the same paper the following question was raised. Question 4.6 ([15]) Let F be a free group of rank n ≥ 3. Is the set of test elements of F negligible? A negative answer to Question 4.6 follows from Theorem 4.1 and the following lemma, which is a slight extension of [6, Proposition 2.1]. Lemma 4.7 Let G be a finitely generated group, X a finite generating set of G, and S ⊆ G. Suppose that G = Sg1 ∪ . . . ∪ Sgm for some g1 , . . . , gm ∈ BX (C). Then ρX (S) ≥ lim inf k→∞

1 |S ∩ BX (k)| ≥ . |BX (k)| m|BX (C)|

Corollary 4.8 ([39]) Let T be the set of test elements of a group G. 1. If G is a free group of rank n ≥ 2, then ρX (T ) ≥ lim inf k→∞

1 |T ∩ BX (k)| ≥ n+1 n |BX (k)| (2 (2 − 1) + 1)|BX (3n − 2)|

for every generating set X of G. 2. If G is a non-orientable surface group of genus n ≥ 3, then ρX (T ) ≥ lim inf k→∞

1 |T ∩ BX (k)| ≥ n−1 |BX (k)| 6 |BX (5n − 5)|

for every generating set X of G. 3. If G is an orientable surface group of genus n ≥ 2, then ρX (T ) ≥ lim inf k→∞

1 |T ∩ BX (k)| ≥ |BX (k)| |BX (161n + 8 · 25n (n − 1)(16n + 1) + 33)|2

for every generating set X of G.

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Since all free groups of finite rank and all surface groups except the fundamental group of the Klein bottle are Turner groups, as a direct consequence we have the following. Corollary 4.9 ([39]) Let R be the union of all proper retracts of a group G. 1. If G is a free group of rank n ≥ 2, then ρX (R) ≤ 1 −

1 (2n+1 (2n − 1) + 1)|BX (3n − 2)|

for every generating set X of G. 2. If G is a non-orientable surface group of genus n ≥ 3, then ρX (R) ≤ 1 −

1 6n−1 |BX (5n − 5)|

for every generating set X of G. 3. If G is an orientable surface group of genus n ≥ 2, then ρX (R) ≤ 1 −

1 |BX (161n + 8 · 25n (n − 1)(16n + 1) + 33)|2

for every generating set X of G. Let F be a free group of rank n ≥ 2 with basis X, and let T and R be, respectively, the set of test elements and the union of all proper retracts of F . It follows from [15, Theorem B] that ρX (T ) ≤ 1 − where ζ(z) =

∞

n=1 n

−z

4n − 4 , (2n − 1)2 ζ(n)

is the Riemann zeta function. Consequently,

ρX (R) ≥ lim inf k→∞

4n − 4 |R ∩ BX (k)| ≥ . |BX (k)| (2n − 1)2 ζ(n)

It follows that T and R have intermediate densities in F . In comparison, it is interesting to note that the union of all proper free factors of F is negligible (cf. [31, Corollary 1.4]).

5 5.1

Concluding remarks and open problems Pro-p groups

Test elements have been studied in various classes of discrete groups (free groups, free nilpotent groups, free solvable groups, Fuchsian groups, torsion free hyperbolic groups, etc.). However, for the category of pro-p groups, not much is known beyond the results surveyed in this paper. It would be interesting to study the test elements of other concrete examples of pro-p groups, especially of the pro-p completions of

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known discrete residually finite-p Turner groups, as by Theorem 3.1, results on test elements of the completions also yield results for the discrete groups. It seems plausible that the notion of a test element has significance for the combinatorial theory of pro-p groups. Some evidence for that is provided by Theorem 2.5. As we already noted, there are examples of pro-p groups for which the conclusion of Theorem 2.5 does not hold. A natural problem is to determine all pro-p groups G which satisfy the following property: for every endomorphism ϕ of G, if ϕ(O(g)) ⊆ O(g) for some g = 1, then ϕ is an automorphism. Some subsets of test elements of a pro-p group exhibit intriguing combinatorial properties. For example, we call an element g of a finitely generated pro-p group G an almost primitive element if g ∈ Φ(G) but g ∈ / Φ(H) whenever H is a proper subgroup of G. It is not difficult to see that every almost primitive element in a finitely generated pro-p group is a test element (see [38, Proposition 5.10]). The set of almost primitive elements of a free pro-p group of finite rank has several pleasant properties (some evidence of this is given in [38, Section 5]). Here we indicate two ways in which almost primitive elements of free pro-p groups occur naturally in the combinatorial theory of pro-p groups. Consider the pro-p group defined by the pro-p presentation G = x, y | xp [y, xp ] . One can show that the relator xp [y, xp ] is not a proper p-th power in the free pro-p group F (x, y) with basis {x, y}. Surprisingly though, this pro-p group also admits the presentation G = x, y | xp . This elegant example discovered by Gildenhuys [12] reveals a sharp disparity between the categories of discrete groups and pro-p groups. (It is a theorem of Magnus (see [24, Theorem 4.12]) that a one-relator discrete group with relator that is not a proper power does not have non-trivial elements of finite order. Moreover, Lyndon [23] proved that such one-relator groups have cohomological dimension at most two.) The following problem, formulated by Gildenhuys, remains open: Let G be a one-relator pro-p group that does not admit a one-relator presentation with a relator that is a proper p-th power. Is the cohomological dimension of G smaller or equal than two? Let r be an almost primitive element of the free pro-p group F (x1 , . . . , xn ) with n ≥ 2. It is easy to see that r is not a proper p-th power in F . Moreover, it can be proved that the group G = x1 , . . . , xn | r does not admit a one-relator presentation with a relator that is a proper p-th power. Therefore, the class of onerelator groups with a relator almost primitive element is a natural testing ground for the problem raised by Gildenhuys. The inner rank of a finitely generated pro-p group G, denoted by Ir(G), is defined to be the maximal rank of a free pro-p group that is an epimorphic image of G. For an element r of a free pro-p group F of finite rank, we set Ir(F, r) := Ir(F/R) where R = r F , the normal closure of r. Let F = F (x1 , ..., xn , y1 , ..., yn , z1 , ..., zm ) αm with be a free pro-p group of rank 2n + m and let r = [x1 , y1 ]...[xn , yn ]z1α1 ...zm 0 = αi ∈ pZp for i = 1, ..., m. In [25], Melnikov proved that Ir(F, r) =  2m+n . 2 Note that r is a test element built from almost primitive elements of free factors of F of ranks 2 and 1 respectively. It turns out that this property of r is the key ingredient in the proof of the above result. In an upcoming paper the authors generalize the result of Melnikov to a wide class of one-relator pro-p groups F/s F

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with a relator s that is a test element of F obtained by combining almost primitive elements of free factors of smaller ranks. 5.2

Discrete groups

Apart from free groups of finite rank and surface groups, we do not know of any other example of an infinite finitely generated discrete group whose set of test elements has intermediate asymptotic density. Moreover, it would be also interesting to find other examples of groups in which the set of test elements forms a net. One major motivation for the study of test elements of a Turner group is the connection between test elements and retracts of the group. For example, we believe that the following holds. Conjecture 5.1 Let H be a finitely generated subgroup of a free group of finite rank F . In addition, suppose that H is not contained in a proper retract of F . Then every test element of H is a test element of F . It can be proved (however, it is not immediate) that this conjecture is equivalent to the following well-known problem on retracts of free groups raised by Bergman [4]. Problem 5.2 Let H be a finitely generated subgroup of a free group of finite rank F , and let R be a retract of F . Is H ∩ R a retract of H? Acknowledgement: This research was partially supported by CNPq. References [1] G. Arzhantseva and A. Olshanskii, Generality of the class of groups in which subgroups with a lesser number of generators are free, (Russian) Mat. Zametki 59 (1996), no. 4, 489–496; translation in: Math. Notes 59 (1996), no. 3-4, 350–355. [2] B. Baumslag, Residually free groups, Proc. London Math. Soc. 17 (1967), 402–418. [3] G. Baumslag, On generalised free products, Math Z. 78 (1962), 423–438. [4] G. M. Bergman, Supports of derivations, free factorizations, and ranks of fixed subgroups in free groups, Trans. Amer. Math. Soc. 351 (1999), 1531–1550. [5] A. V. Borovik, A. G. Myasnikov and V. Shpilrain, Measuring Sets in Infinite Groups, Computational and statistical group theory (Las Vegas, NV/Hoboken, NJ, 2001), 21-42, Contemp. Math., 298, Amer. Math. Soc., Providence, RI, 2002. [6] J. Burillo and E. Ventura, Counting primitive elements in free groups, Geom. Dedicata 93 (2002), 143–162. [7] S. Demushkin, The group of a maximal p-extension of a local field, Izv. Akad. Nauk, USSR Math. Ser. 25 (1961), 329–346. [8] S. Demushkin, On 2-extensions of a local field, Sibirsk. Mat. Z. 4 (1963), 951–955. [9] J. D. Dixon, M. P. F. du Sautoy, A. Mann, and D. Segal, Analytic pro-p groups, Cambridge Studies in Advanced Mathematics 61, Cambridge University Press, Cambridge, second edition, 1999. [10] B. Fine, G. Rosenberger, D. Spellman and M. Stille, Test words, generic elements and almost primitivity, Pac. J. Math. 190 (1999), 277–297.

500

Snopce, Tanushevski: Test elements: from pro-p to discrete groups

[11] S. V. Ivanov, On endomorphisms of free groups that preserve primitivity, Arch. Math. 72 (1999), 92–100. [12] D. Gildenhuys, On pro-p groups with a single defining relator, Invent. Math. 5 (1968), 357–366. [13] R. Z. Goldstein, The density of small words in a free group is 0, Group theory, statistics, and cryptography, 47–50, Contemp. Math., Amer. Math. Soc., 360, Providence, RI, 2004. [14] D. Groves, Test elements in torsion-free hyperbolic groups, New York J. Math. 18 (2012), 651–656. [15] I. Kapovich, I. Rivin, P. Schupp and V. Shpilrain, Densities in free groups and Zk , visible points and test elements, Math. Research Letters, 14 (2007), no.2, 263–284. [16] I. Kapovich, P. Schupp and V. Shpilrain, Generic properties of Whiteheads Algorithm and isomorphism rigidity of random one-relator groups, Pacific J. Math. 223 (2006), no. 1, 113–140. [17] G. Klaas, C. R. Leedham-Green and W. Plesken, Linear pro-p groups of finite width. Lecture Notes in Mathematics, 1674. Springer-Verlag, Berlin, 1997. [18] J. P. Labute, Classification of Demushkin groups, Canad. J. Math. 19 (1967), 106–132. [19] D. Lee, Primitivity preserving endomorphisms of free groups, Commun. Algebra 30 (2002), 1921–1947. [20] D. Lee, Endomorphisms of free groups that preserve automorphic orbits, J. Algebra 248 (2002), 230–236. [21] H. Leptin, Ein Darstellungssatz f¨ ur kompakte total unzusammenh¨ angende Gruppen, Arch. Math. 6 (1955), 371–373. [22] G. Levitt and A. Minasyan, Residual properties of automorphism groups of (relatively) hyperbolic groups, Geom. Topol. 18 (2014) 2985–3023. [23] R. Lyndon, Cohomology theory of groups with a single defining relation, Ann. Math. 52 (1950), 650–665. [24] W. Magnus, A. Karrass and D. Solitar, Combinatorial group theory. Wiley, New York, 1966. [25] O. V, Mel’nikov, Products of powers and commutators in free pro-p groups, Mat. Zametki 40 (1986), 433–441. [26] A. Mikhalev, V. Shpilrain and J. T. Yu, Combinatorial Methods. Free groups, polynomials and free algebras. CMS Books in Mathematics/ Ouvrages de Math´ematiques de la SMC, 19. Springer-Verlag, New York, 2004. [27] J. Neukirch, A. Schmidt and K. Wingberg, Cohomology of Number Fields. Second edition. Springer-Verlag, Berlin (2008). [28] J. Nielsen, Die Isomorphismen der allgemeinen, unendlichen Gruppe mit zwei Erzeugenden. Math. Ann. 79 (1918), 385–397. [29] N. Nikolov and D. Segal, On finitely generated profinite groups, I: strong completeness and uniform bounds, Ann. of Math. 165 (2007), 171–238. [30] J. O’Neill and E. C. Turner, Test elements and the retract theorem in hyperbolic groups, New York J. Math. 6 (2000), 107–117. [31] D. Puder and C. Wu, Growth of primitive elements in free groups, J. London Math. Soc. 90 (2014), no. 1, 89–104. [32] L. Ribes and P. Zalesskii, Profinite groups. 2nd edition. Springer-Verlag, Berlin (2010). [33] E. Rips, Commutator equations in free groups, Israel J. Math. 39 (1981), 326–340. [34] J. P. Serre, Structure de certains pro-p groupes, S´eminaire Bourbaki 1962/63, no. 252 (1971), 357–364. [35] J. P. Serre, Galois Cohomology. Springer-Verlag, Berlin (2002).

Snopce, Tanushevski: Test elements: from pro-p to discrete groups

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[36] V. Shpilrain, Recognizing automorphisms of the free group, Arch. Math. 62 (1994), 385–392. [37] V. Shpilrain, Generalized primitive elements of a free group, Arch. Math. 71 (1998), 270–278. [38] I. Snopce, S. Tanushevski, Test elements in pro-p groups with applications in discrete groups, Israel J. Math. 219 (2017), 783–816. [39] I. Snopce, S. Tanushevski, Asymptotic density of test elements in free groups and surface groups, Int. Math. Res. Not. 2017 (2017), 5577–5590. [40] I. Snopce, S. Tanushevski, Products of test elements of free factors of a free group, J. Algebra, 474 (2017), 271-287. [41] J. Sonn, Epimorphisms of Demushkin groups, Israel J. Math. 17 (1974), 176–190. [42] J. R. Stallings, Whitehead graphs on handlebodies. Geometric group theory down under (Canberra, 1996), 317–330, de Gruyter, Berlin, 1999. [43] E. C. Turner, Test words for automorphisms of free groups, Bull. London Math. Soc. 28 (1996), 255–263. [44] J. S. Wilson, Profinite groups, The Clarendon Press, New York, 1998. [45] J. S. Wilson, On abstract and profinite just infinite groups. In: New horizons in pro-p groups. Progr. Math. vol. 184, pp. 181–203. Birkhauser, Boston (2000). [46] W. Woess, Cogrowth of groups and simple random walks, Arch. Math. 41 (1983), 363–370. [47] H. Zieschang, Alternierende Produkte in freien Gruppen, Abh. Math. Sem. Univ. Hamburg 27 (1964), 12–31. [48] H. Zieschang, Uber Automorphismen ebener discontinuerlicher Gruppen, Math. Ann. 166 (1966), 148–167.